2. Key ideas, in comparison with previous work

By now there are several different proofs of the $O(1/\sqrt{n})$ bound in the (linear) Erdős–Littlewood–Offord theorem. As far as we know, they all take advantage of at least one of two very special properties of random variables of the form $X=a_{1}\xi_{1}+\dots+a_{n}\xi_{n}$ . First, Erdős’ original proof [Reference ErdősErd45] used the monotonicity of X (for every index i, changing $\xi_{i}$ from $-1$ to 1 will always make the value of X increase, or always make the value of X decrease). Second, one can take advantage of the fact that X is a sum of independent random variables (each with a very simple distribution), so its Fourier transform is very well behaved. (The first Fourier-analytic proof seems to have been by Halász [Reference HalászHal77]; see also the very simple proof of a $O(1/\sqrt n)$ bound due to Croot [Reference CrootCro11].)

Unfortunately, in the quadratic setting (where $X=Q(\xi_{1},\ldots,\xi_{n})$ for some quadratic polynomial Q), both of the above properties of X may fail in a very strong way. There are two general approaches that have been most successful so far: Gaussian approximation, and a technique called decoupling. We briefly discuss both these approaches and their limitations, before describing our new ideas.

2.2 Decoupling

A completely different technique was employed in the papers of Costello, Tao and Vu [Reference Costello, Tao and VuCTV06] and Rosński and Samorodnitsky [Reference Rosiński and SamorodnitskyRS96], which first studied the quadratic Littlewood–Offord problem. This technique is now usually called decoupling, following [Reference Costello, Tao and VuCTV06]. Roughly speaking, decoupling is a general technique to ‘reduce a quadratic anticoncentration problem to a linear one’.Footnote ⁶ Below, we sketch the basic idea, incorporating an improvement due to Costello and Vu [Reference Costello and VuCV08].Footnote ⁷

Let $Q\in \mathbb{R}[x_1,\ldots,x_n]$ be a quadratic polynomial and let $\vec{\xi}=(\xi_{1},\ldots,\xi_{n})$ be a sequence of independent Rademacher random variables. Given a partition of the index set $\{1,\ldots,n\}$ into two subsets I and J, we can break the random vector $\vec{\xi}$ into two parts $\vec{\xi}[I]\in\{-1,1\}^{I}$ and $\vec{\xi}[J]\in\{-1,1\}^{J}$ . Then, a simple application of the Cauchy–Schwarz inequality (see Lemma 3.6) shows that, if $\vec{\xi}\mkern2mu\vphantom{\xi}'[I]$ is an independent copy of $\vec{\xi}[I]$ , we have

(2.1) \begin{align}\mathrm{Pr}[Q(\vec \xi)=z]=\mathrm{Pr}[Q(\vec{\xi}[I],\vec{\xi}[J])=z] & \le\mathrm{Pr}[Q(\vec{\xi}[I],\vec{\xi}[J])=z \text{ and } Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec{\xi}[J])=z]^{1/2}\end{align}

(2.2) \begin{align}& \le \mathrm{Pr}[Q(\vec{\xi}[I],\vec{\xi}[J])-Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec{\xi}[J])=0]^{1/2}.\end{align}

Now, $Q(\vec{\xi}[I],\vec{\xi}[J])-Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec{\xi}[J])$ can be interpreted as a linear function of $\vec{\xi}[J]$ , with coefficients that depend on $(\vec{\xi}[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I])$ (since the terms that are quadratic in $\vec \xi[J]$ cancel out). Furthermore, this linear function typically has many nonzero coefficients (since it is unlikely that most of the ‘cross-terms’ between $\vec \xi[I]$ and $\vec{\xi}\mkern2mu\vphantom{\xi}'[I]$ cancel out). So, after conditioning on a typical outcome of $(\vec{\xi}[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I])$ , one can apply the Erdős–Littlewood–Offord theorem to obtain a bound of the form $\mathrm{Pr}[Q(\vec \xi)=z]\le (O(1/\sqrt n))^{1/2}=O(n^{-1/4})$ .

The great advantage of decoupling is that the resulting linear anticoncentration problem is much easier: one has access to the much wider range of tools available to study sums of independent random variables. However, the inequality between Equations (2.1) and (2.2) is rather lossy, and one tends to end up with bounds that are at least ‘a square root away’ from best-possible.Footnote ⁸

Costello’s paper [Reference CostelloCos13], proving the first bound of the form $n^{-1/2+o(1)}$ for the quadratic Littlewood–Offord problem, combined decoupling with a structural dichotomy. Namely, Costello discovered that if the quadratic polynomial Q is in a certain sense ‘robustly irreducible’, then number-theoretic arguments give the stronger bound $\mathrm{Pr}[Q(\vec{\xi}[I],\vec{\xi}[J])-Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec{\xi}[J])=0]\le n^{-1+o(1)}$ , and so, even after the ‘square-root loss’ of decoupling, one has $\mathrm{Pr}[Q(\vec{\xi}[I],\vec{\xi}[J])=z]\le n^{-1/2+o(1)}$ . He then gave a separate argument to handle the case where Q does not satisfy the robust irreducibility condition (i.e., when Q essentially splits into linear factors), based on the Szemerédi–Trotter theorem from discrete geometry.

We cannot conclusively rule out the possibility that one could prove an optimal $O(1/\sqrt n)$ bound with a similar kind of case analysis, but this seems to be extremely difficult. In particular, we are not aware of any suitable candidate for a condition on Q which ensures that $\mathrm{Pr}[Q(\vec{\xi}[I],\vec{\xi}[J])-Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec{\xi}[J])=0]\le O(1/n)$ (Costello’s notion of robust irreducibility, and similar notions of ‘robust rank’ or ‘robust sum-of-squares complexity’, are only suitable for an $n^{-1+o(1)}$ bound on such probabilities). Also, Costello’s Szemerédi–Trotter-based proof for the nearly reducible case does not seem to easily generalise to other simple-looking families of quadratic polynomials (for example, polynomials which can be written as the sum of four squares; see the discussion in [Reference Fox, Kwan and SpinkFKS23, Section 10]).

2.3 A geometric point of view, and an inductive decoupling scheme

In this paper, we consider a different perspective on decoupling: instead of using decoupling to immediately reduce from a quadratic problem to a linear one, we reinterpret decoupling as a tool to obtain a problem with a linear and a quadratic part, and to inductively ‘reduce the proportion of our problem that is quadratic’. To explain this, it is helpful to take a geometric perspective.

Specifically, for a quadratic polynomial $Q\in \mathbb{R}[x_1,\ldots,x_n]$ and a random vector $\vec{\xi}\in \{1,-1\}^n$ , note that $\mathrm{Pr}[Q(\vec \xi)=z]$ can be interpreted as the probability that $\vec \xi$ lies in the quadric (quadratic variety) $\mathcal Z$ given by $\mathcal Z=\{\vec x \in \mathbb{R}^n: Q(\vec x)=z\}\subseteq\mathbb{R}^n$ . Similarly, the expression $\mathrm{Pr}[Q(\vec{\xi}[I],\vec{\xi}[J])=z\text{ and }Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec{\xi}[J])=z]$ appearing in Equation (2.1) can be interpreted as the probability that $\vec \xi[J]$ lies in the variety

$$ \mathcal Z^{(1)}=\{\vec x\in \mathbb{R}^J: Q(\vec{\xi}[I],\vec x)=z \text{ and } Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec x)=z\}=\mathcal Z_{\vec \xi[I]}\cap \mathcal Z_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}\subseteq \mathbb{R}^J, $$

where, for $\vec u\in \mathbb{R}^I$ , we write $\mathcal Z_{\vec u}$ for the set of all $\vec x\in \mathbb{R}^J$ with $(\vec u,\vec x)\in \mathcal Z$ (typically, this is a quadric in $\mathbb{R}^J$ ). Using this language, Equation (2.1) can be restated as

(2.3) \begin{equation}\mathrm{Pr}[Q(\vec \xi)=z]=\mathrm{Pr}[\vec \xi\in \mathcal Z]\le \mathrm{Pr}[\vec \xi[J]\in \mathcal Z^{(1)}]^{1/2}.\end{equation}

The next step leading to the traditional decoupling inequality, Equation (2.2), can geometrically be phrased as observing that $\mathcal Z^{(1)}$ lies inside the affine-linear subspace

$$ \mathcal W^{(1)}=\{\vec x \in \mathbb{R}^J :Q(\vec{\xi}[I],\vec{x})-Q(\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec{x})=0\}. $$

Indeed, this yields the probability bound

$$ \mathrm{Pr}[Q(\vec \xi)=z]\le \mathrm{Pr}[\vec \xi[J]\in \mathcal Z^{(1)}]^{1/2}\le \mathrm{Pr}[\vec \xi[J]\in \mathcal W^{(1)}]^{1/2} $$

stated in Equation (2.2). One can then forget about $\mathcal Z^{(1)}$ and restrict one’s attention to the affine-linear subspace $\mathcal W^{(1)}$ , where the relevant probabilities are easier to analyse (under suitable assumptions on Q, one can show that $\mathrm{Pr}[\vec \xi[J]\in \mathcal W^{(1)}]\le O(1/\sqrt{n})$ , leading to the bound $\mathrm{Pr}[Q(\vec \xi)=z]\le O(n^{-1/4})$ described in § 2.2).

However, it turns out that this ‘forgetting’ of $\mathcal{Z}^{(1)}$ is precisely the cause of the square-root loss usually associated with decoupling. Indeed, $\mathcal W^{(1)}$ is a variety with codimension 1 (being defined by a single equation), while $\mathcal{Z}^{(1)}$ is typicallyFootnote ⁹ a variety with codimension (at least) 2 (being defined by two equations), so intuitively we should be much less likely to have $\vec{\xi}[J]\in\mathcal{Z}^{(1)}$ than $\vec{\xi}[J]\in\mathcal{W}^{(1)}$ . More specifically, in order to have $\vec{\xi}[J]\in\mathcal{Z}^{(1)}$ , we need $\vec{\xi}[J]$ to satisfy two different equations simultaneously, and we might expect each of these to be satisfied with probability $O(1/\sqrt n)$ . So for typical outcomes of $\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]$ (which determine $\mathcal Z^{(1)}$ ) we might expect $\mathrm{Pr}[\vec{\xi}[J]\in\mathcal{Z}^{(1)}\,|\,\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]]\le(O(1/\sqrt{n}))^{2}$ . If we could prove this, we would be able to deduce Theorem 1.1 (recalling Equation (2.3)).

So, we choose not to ‘forget’ $\mathcal{Z}^{(1)}$ , and our task is to show that $\mathrm{Pr}[\vec{\xi}[J]\in\mathcal{Z}^{(1)}]\le(O(1/\sqrt{n}))^{2}$ . While this new task may seem harder than the previous one (as $\mathcal{Z}^{(1)}$ seems like a more complicated object than $\mathcal{Z}$ ), the key observation is that we have ‘reduced the relative proportion of the quadratic part of our problem’. Indeed, at the start we were interested in a quadric $\mathcal{Z}$ described by a single quadratic equation, but now we are interested in $\mathcal{Z}^{(1)}$ , which can be interpreted as a quadric inside the affine-linear subspace $\mathcal{W}^{(1)}\subseteq \mathbb{R}^J$ . That is to say, $\mathcal{Z}^{(1)}$ is described by one linear and one quadratic equation, so now ‘only half of our problem is quadratic’.

Crucially, it is possible to iterate this entire procedure: we next fix a partition $I^{(2)}\cup J^{(2)}$ of J, and consider the variety

$$ \mathcal{Z}^{(2)}=\mathcal{Z}^{(1)}_{\vec \xi[I^{(2)}]}\cap\mathcal{Z}^{(1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]}\subseteq \mathbb{R}^{J^{(2)}}; $$

(where, for $\vec u\in \mathbb{R}^{I^{(2)}}$ , we write $\mathcal Z^{(1)}_{\vec u}$ for the set of all $\vec x\in \mathbb{R}^{J^{(2)}}$ with $(\vec u,\vec x)\in \mathcal Z^{(1)}$ ). Now, decoupling, analogously to the inequality in Equation (2.3), yields

(2.4) \begin{equation}\mathrm{Pr}[\vec \xi[J]\in \mathcal Z^{(1)}\,|\,\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]]\le \mathrm{Pr}[\vec \xi[J^{(2)}]\in \mathcal Z^{(2)}\,|\,\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]]^{1/2}.\end{equation}

So, it suffices to show that $\mathrm{Pr}[\vec \xi[J^{(2)}]\in \mathcal Z^{(2)}\,|\,\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]]\le (O(1/\sqrt n))^4$ for typical outcomes of $\vec \xi[I]$ and $\vec{\xi}\mkern2mu\vphantom{\xi}'[I]$ . Now, for $\mathcal{Z}^{(2)}$ to be nonempty, it must be the case that $\mathcal{W}^{(1)}_{\vec \xi[I^{(2)}]}\cap\mathcal{W}^{(1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]}$ is nonempty, or equivalently that $\mathcal{W}^{(1)}_{\vec \xi[I^{(2)}]}=\mathcal{W}^{(1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]}$ (it is not hard to see that $\mathcal{W}^{(1)}_{\vec \xi[I^{(2)}]}$ and $\mathcal{W}^{(1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]}$ are parallel translates of each other). That is to say, $\vec \xi[I^{(2)}]-\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]$ must lie in a certain affine-linear subspace which typically has codimension 1; this happens with probability $O(1/\sqrt n)$ by the (linear) Erdős–Littlewood–Offord theorem.

If we also condition on outcomes of $\vec \xi[I^{(2)}]$ and $\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]$ such that $\mathcal{W}^{(1)}_{\vec \xi[I^{(2)}]}=\mathcal{W}^{(1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]}$ , it is not hard to see that $\mathcal Z^{(2)}$ is typically a quadric inside the affine-linear subspace $\mathcal W^{(2)}$ of $\mathcal{W}^{(1)}_{\vec \xi[I^{(2)}]}=\mathcal{W}^{(1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]}\subseteq \mathbb{R}^{J^{(2)}}$ given by the linear equation $Q(\vec \xi[I],\vec \xi[I^{(2)}],\vec x)-Q(\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}],\vec x)=0$ (in much the same way that $\mathcal Z^{(1)}$ is a quadric inside the affine-linear subspace $\mathcal W^{(1)}$ ). That is to say, $\mathcal Z^{(2)}$ is typically a variety of codimension (at least) 3, described by two linear equations and one quadratic equation. So we might expect that (for typical outcomes of $\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec \xi[I^{(2)}],\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]$ for which $\mathcal{W}^{(1)}_{\vec \xi[I^{(2)}]}=\mathcal{W}^{(1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]}$ )

(2.5) \begin{equation}\mathrm{Pr}[\vec \xi[J^{(2)}]\in\mathcal{Z}^{(2)}\,|\,\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec \xi[I^{(2)}],\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]]\le(O(1/\sqrt{n}))^{3}.\end{equation}

If we were able to prove Equation (2.5), we would obtain a bound of the form

$$ \mathrm{Pr}[\vec \xi[J^{(2)}]\in \mathcal Z^{(2)}]\le O(1/\sqrt{n})\cdot (O(1/\sqrt{n}))^{3}=(O(1/\sqrt{n}))^{4}, $$

which would imply Theorem 1.1, tracing back through our decoupling inequalities Equation (2.3) and Equation (2.4). We have made progress by ‘reducing the proportion of our problem that is quadratic’: if, instead of Equation (2.5), we were only able to prove that

\begin{align*}\mathrm{Pr}[\vec \xi[J^{(2)}]\in\mathcal{Z}^{(2)}\,|\,\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec \xi[I^{(2)}],\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]] & \le \mathrm{Pr}[\vec \xi[J^{(2)}]\in\mathcal{W}^{(2)}\,|\,\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I],\vec \xi[I^{(2)}],\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(2)}]] \\& \le(O(1/\sqrt{n}))^{2} \end{align*}

(‘forgetting’ the quadratic part of the problem and only bounding the probability that $\vec \xi[J^{(2)}]$ lies in the affine-linear subspace $\mathcal{W}^{(2)}$ of codimension 2), we would end up with a final bound of the form $\mathrm{Pr}[\vec \xi\in \mathcal Z]\le O(n^{-3/8})$ , which is much better than the $O(n^{-1/4})$ bound we obtained with a single decoupling step.

In general, after k steps of this scheme, we will have considered k ‘nested’ partitions of the form $J^{(i-1)}=I^{(i)}\cup J^{(i)}$ , and defined k varieties $\mathcal Z^{(1)},\ldots,\mathcal Z^{(k)}$ . We will have applied the decoupling inequality k times, and considered various conditional probabilities that the vectors $\vec \xi[I^{(i)}]-\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(i)}]$ lie in certain affine-linear subspaces (to ensure that certain intersections $\mathcal{W}^{(i-1)}_{\vec \xi[I^{(i)}]}\cap\mathcal{W}^{(i-1)}_{\vec{\xi}\mkern2mu\vphantom{\xi}'[I^{(i)}]}$ are nonempty). After all this, we find ourselves in a position where if we ‘forget the quadratic part of the problem’ we obtain a bound of the form $\mathrm{Pr}[Q(\vec \xi)=z]\le O(n^{-1/2+1/2^{k+1}})$ . That is to say, if k is at least $\log \log n$ , the quadratic part of the problem is so insignificant that ‘forgetting’ it only costs us a constant factor in the final bound.

At an extremely high level, this explains our strategy to prove Theorem 1.1. However, we omitted a number of important details in this outline. In particular, our strategy heavily depends on being able to obtain suitable upper bounds on probabilities that certain random vectors fall into certain affine-linear subspaces, and there are crucial ‘robust rank’ nondegeneracy conditions that must be satisfied in order for such bounds to hold. This is not just a technicality; we require significant new ideas to maintain these robust rank properties during our iterative decoupling scheme, which we discuss in the next subsection.

Remark 2.1. In the actual proof, in order to keep everything organized, we phrase our arguments recursively: the key step is to prove a recursive bound for the maximum possible probability that a random vector $\vec{\xi}\in \{1,-1\}^n$ falls onto some quadric on an affine-linear subspace of a given codimension, satisfying certain nondegeneracy conditions (this main recursive bound is stated in Theorem 5.5).

Remark 2.2. Theorem 1.1 concerns quadratic polynomials, but one may be interested in a generalisation to cubic polynomials, or to polynomials of any fixed degree. Decoupling still makes sense for general polynomials: for example, if $\mathcal Z$ is a cubic (variety), then decoupling yields an inequality of the form $\mathrm{Pr}[\vec \xi\in \mathcal Z]\le \mathrm{Pr}[\vec\xi[J]\in \mathcal Z^{(2)}]^{1/2}$ , where $\mathcal Z^{(2)}$ is the intersection of a quadric and a cubic. As observed by Rosiński and Samorodnitsky [Reference Rosiński and SamorodnitskyRS96] and Razborov and Viola [Reference Razborov and ViolaRV13], the basic type of decoupling argument described in § 2.2 generalises quite straightforwardly to higher degrees (but the bounds get worse as the degree increases). However, it is less clear how to generalise the inductive decoupling scheme described in this subsection. Roughly speaking, in the degree-d case, instead of affine-linear subspaces (obtained as intersections of affine-linear hyperplanes), one must work with intersections of degree- $(d-1)$ varieties, which are much more complicated objects. We hope that the relevant complexities can be handled with some kind of multiple-level induction, but so far we were not able to accomplish this.

2.4 High-dimensional anticoncentration inequalities and witness counting

Our proof, as outlined in the previous subsection, relies on bounds on probabilities that random vectors lie in certain affine-linear subspaces. More specifically, for a suitably nondegenerate affine-linear subspace $\mathcal W\subseteq \mathbb{R}^n$ of codimension k, and a uniformly random vector $\vec{\xi}\in \{1,-1\}^n$ , we need a probability bound of the form $\mathrm{Pr}[\vec \xi\in \mathcal W]\le O(n^{-k/2})$ . Intuitively, this is because $\vec \xi$ needs to simultaneously satisfy k different linear equations, each of which is satisfied with probability roughly $n^{-1/2}$ . More formally, such a bound follows from a high-dimensional version of the Erdős–Littlewood–Offord theorem.

The first such high-dimensional version was due to Halász [Reference HalászHal77]. In linear-algebraic language, it can be phrased as follows: for any fixed k, if $M\in \mathbb R^{k\times n}$ is a matrix that ‘robustly has rank k’ in the sense that (for some fixed $\delta \gt 0$ ) one cannot delete $\delta n$ columns of M to obtain a matrix with rank less than k, then for a uniformly random vector $\vec \xi\in\{-1,1\}^n$ we have

$$ \sup_{\vec w\in \mathbb R^k}\mathrm{Pr}[M\vec \xi=\vec w]\le O(n^{-k/2}). $$

Note that some kind of ‘robust rank-k’ condition is necessary here: for example, if n is even and $M\in \mathbb{R}^{2\times n}$ has rows $(1,\ldots,1)\in \mathbb{R}^n$ and $(1,\ldots,1,0,0)\in \mathbb{R}^n$ , then it is easy to check that $\mathrm{Pr}[M\vec \xi=\vec 0]$ has order of magnitude $1/\sqrt n$ .

Several extensions and variants of Halasz’ inequality have since been proved (see for example [Reference Ferber, Jain and ZhaoFJZ22, Reference Fox, Kwan and SauermannFKS21, Reference HowardHow00, Reference Tao and VuTV06]); in particular, Ferber, Jain and Zhao [Reference Ferber, Jain and ZhaoFJZ22] proved a version of Halasz’ theorem with a much better dependence on k (allowing k to vary with n, instead of viewing it as a constant). We state (a corollary of) this theorem as Theorem 3.3.

Of course, whenever we want to apply any Halász-type theorem, we need a ‘robust rank’ condition to hold. So, in order to execute the strategy described in the last subsection, at each step of the decoupling scheme we need a ‘robust rank inheritance’ lemma, proving that a robust rank condition is likely to hold for the next step, given that it holds for the current step. The key ingredient for our robust rank inheritance lemma is a new high-dimensional anticoncentration inequality for the probability that a random vector falls in a small ball in the Hamming norm. We believe this inequality (and the techniques in its proof) to be of independent interest; an important special case is as follows.

Lemma 2.1. For any fixed positive integer r, there are constants $C_r \gt 0$ and $c_r \gt 0$ only depending on r such that the following holds. Consider a matrix $A\in \mathbb{R} ^{m\times n}$ which has rank at least r after deletion of any t rows and t columns (for some positive integer t). Then for a sequence $\vec{\xi}=(\xi_{1},\ldots,\xi_{n})\in\{-1,1\}^{n}$ of independent Rademacher random variables, we have

$$\sup_{\vec v\in \mathbb R^m}\mathrm{Pr}[A\vec{\xi}\text{ differs from }\vec{v}\text{ in fewer than }c_r t\text{ coordinates}]\le C_r \cdot t^{-r/2}. $$

The assumption in Lemma 2.1 says that A robustly has rank at least r, but in a stronger sense than typical Halász-type theorems: the rank needs to remain at least r after row deletion as well as column deletion. As a result, we are able to obtain a stronger conclusion, namely that for any vector $\vec v$ , it is unlikely that $A\vec \xi$ agrees with $\vec v$ in almost all its coordinates. (It is not hard to see that for this stronger conclusion, such a row-deletion assumption is indeed necessary.)

We prove Lemma 2.1 (and our more general robust rank inheritance lemma) in § 7 using a witness-counting technique, which we outline here. First, note that the most naïve strategy to prove Lemma 2.1 would be to simply take a union bound over all sets I of $m-c_r t$ coordinates in which $A\vec \xi$ and $\vec v$ could agree. For some specific I, the probability that $A\vec \xi$ and $\vec v$ agree on the coordinates indexed by I can be easily understood using existing tools (i.e., Halász’ inequality and its variants) and is at most of the order of $t^{-r/2}$ . However, it is far too wasteful to simply take a union bound summing over all possibilities for I (the number of possibilities is exponential in t).

Instead, for each I we consider small ‘witness’ subsets $I'\subseteq I$ , such that the submatrix of A consisting of the rows with indices in I’ still has (robustly) high rank. Note that for each $I'\subseteq I$ , whenever $A\vec \xi$ and $\vec v$ agree on the coordinates indexed by I, they also in particular agree on the coordinates indexed by I’. Given the high-rank property, for each ‘witness’ subset I’ we can still easily bound the probability that $A\vec \xi$ and $\vec v$ agree on the coordinates indexed by I’ (using Halász’ inequality and its variants).

There are still too many possible ‘witness’ subsets I’ to be able to simply take a union bound over all possibilities for I’, but (roughly speaking) we can show that whenever $A\vec \xi$ and $\vec v$ agree on a large set of coordinates I, then they must agree on the coordinates of many ‘witness’ subsets $I'\subseteq I$ . We can show this is unlikely by computing the expected number of ‘witness’ coordinate subsets on which $A\vec \xi$ and $\vec v$ agree, and applying Markov’s inequality.

Remark 2.3. One might be interested in adapting Theorem 1.1 to study small-ball probabilities of the form $\sup_{z\in \mathbb{R}}\mathrm{Pr}[|Q(\xi_{1},\ldots,\xi_{n})-z|\le 1]$ instead of point probabilities $\sup_{z\in \mathbb{R}}\mathrm{Pr}[Q(\xi_{1},\ldots,\xi_{n})=z]$ . The robust rank inheritance lemma seems to be the main point of difficulty for such an adaptation; it is not clear how to extend the witness-counting arguments to the small-ball setting (e.g., in the setting of Lemma 2.1, we would be interested in the event that there are fewer than $c_r t$ coordinates i for which $|(A\vec \xi-\vec v)[i]|\le 1$ ).

Before ending this overview section, we remark that there is a way to sidestep the robust rank inheritance issue in the special case where Q has ‘bounded rank’, meaning that the quadratic part of Q can be written as $\vec x^{\intercal} A\vec x$ for some symmetric matrix A of rank O(1). Indeed, in this case we can reduce our entire problem to a certain bounded-dimensional geometric anticoncentration problem (involving a quadric), where the robust rank conditions for Halasz’ inequality are always automatically satisfied when following the strategy in the previous subsection. In § 4, we give a simple self-contained proof of an essentially optimal anticoncentration bound in this setting. We believe this to be a good illustration of the basic principles of our inductive decoupling scheme, with minimal technicalities. Moreover, the result of § 4 will actually also be an ingredient for the full proof of Theorem 5.1: it allows us to restrict our attention to the case where the quadratic part of Q ‘robustly has high rank’, which allows us to avoid possible degeneracies at certain points in the proof.

3. Preliminaries

First, as outlined in § 2, we need a ‘high-dimensional’ version of the Erdős–Littlewood–Offord theorem. This will require a robust rank condition, defined as follows.

We clearly have $\mathcal{H}^{k\times n}(s)\subseteq \mathcal{H}^{k\times n}(s')$ if $s\ge s'$ . Furthermore, for a partition $[n]=S\cup I$ with $|S|\le s$ , for every matrix $M\in \mathcal{H}^{k\times n}(s)$ we also have $M[[k]\times I]\in \mathcal{H}^{k\times I}(s-|S|)$ . Also note that in the case $k=0$ , the (unique) empty matrix $M\in\mathbb{R}^{0\times n}$ is contained in $\mathcal{H}^{0\times n}(s)$ for all $s\ge 0$ .

For integers $0\le k\le n$ and $t\ge 0$ , we say that a matrix $M\in \mathbb{R}^{k\times n}$ contains t disjoint nonsingular $k\times k$ submatrices if there exist disjoint subsets $J_1,\ldots, J_t\subseteq [n]$ of size $|J_1|=\dots=|J_t|=k$ such that $\operatorname{rank} M[[k]\times J_i]=k$ for $i=1,\ldots,t$ . This is the case if and only if among the column vectors $\vec{a}_1,\ldots,\vec{a}_n$ of M we can form t disjoint bases of $\mathbb{R}^{k}$ (here, the vectors $\vec{a}_1,\ldots,\vec{a}_n$ are considered with multiplicities, i.e., a vector in $\mathbb{R}^{k}$ can occur in two different bases if it occurs twice among $\vec{a}_1,\ldots,\vec{a}_n$ ).

1. (i) if $M\in \mathbb{R}^{k\times n}$ contains t disjoint nonsingular $k\times k$ submatrices for an integer $t \gt s$ , then $M\in \mathcal{H}^{k\times n}(s)$ ;

2. (ii) if $M\in \mathcal{H}^{k\times n}(s)$ and $k\ge 1$ , then M contains $\lceil s/k\rceil$ disjoint nonsingular $k\times k$ submatrices.

For (ii), we can greedily find disjoint subsets $J_1,\ldots, J_{\lceil s/k\rceil}\subseteq [n]$ of size $|J_1|=\dots=|J_{\lceil s/k\rceil}|=k$ such that $\operatorname{rank} M[[k]\times J_i]=k$ for $i=1,\ldots,\lceil s/k\rceil$ . Indeed, after having chosen $J_1,\ldots, J_{i-1}$ for some $i\le \lceil s/k\rceil$ , we have $|J_1\cup \dots\cup J_{i-1}|=(i-1)k<s$ and hence M still has rank k after deleting the columns with indices in $J_1\cup \dots\cup J_{i-1}$ . So we can find a subset $J_i\subseteq [n]\setminus (J_1\cup \dots\cup J_{i-1})$ of size $|J_i|=k$ with $\operatorname{rank} M[[k]\times J_i]=k$ .

Halász [Reference HalászHal77] proved that for fixed constants $\delta \gt 0$ and $k\in \mathbb{N}$ , if $M\in \mathcal{H}^{k\times n}(\delta n)$ then $\mathrm{Pr}[M\vec \xi=\vec w]\le O(n^{-k/2})$ . We will use the following quantitative version of Halász’ theorem, which is a special caseFootnote ¹⁰ of a result of Ferber, Jain and Zhao [Reference Ferber, Jain and ZhaoFJZ22, Theorem1.11] (see also [Reference HowardHow00] for a previous result with a weaker dependence on k).

Theorem 3.3. Let $0\le k\le n$ and $t\ge 1$ be integers. Consider a matrix $M\in \mathbb{R}^{k\times n}$ containing t disjoint nonsingular $k\times k$ submatrices, and a vector $\vec{w}\in \mathbb{R}^k$ . Letting $\vec{\xi}=(\xi_{1},\ldots,\xi_{n})\in\{-1,1\}^n$ be a sequence of independent Rademacher random variables, we then have

$$ \mathbb{P}[M\vec{\xi}=\vec{w}]\le t^{-k/2}. $$

Remark 3.1. The anonymous referee brought to our attention that Theorem 3.3 is essentially the only aspect of this paper which is not self-contained, and therefore it seems worthwhile to briefly sketch its short proof. (Of course, the reader may still wish to refer to [Reference Ferber, Jain and ZhaoFJZ22] for the full details of the proof.) Let $M_1,\ldots,M_t\in \mathbb R^{k\times k}$ be disjoint nonsingular $k\times k$ submatrices of M. First, it is not hard to see that, for any $\ell\le t$ , we have

$$ \sup_{\vec w\in \mathbb R^k}\mathrm{Pr}[M\vec{\xi}=\vec{w}]\le \sup_{\vec z\in \mathbb R^k}\mathrm{Pr}[M_1\vec{\xi}^{\,(1)}+\dots +M_\ell\vec{\xi}^{\,(\ell)}=\vec z], $$

where $\vec \xi^{\,(1)},\ldots,\vec\xi^{\,(\ell)}\in \{-1,1\}^k$ are independent sequences each consisting of k independent Rademacher random variables. Then, the key step is a ‘replication trick’, showing that if $\ell$ is even, we have

$$ \sup_{\vec z\in \mathbb R^k}\mathrm{Pr}[M_1\vec{\xi}^{\,(1)}+\dots +M_\ell\vec{\xi}^{\,(\ell)}=\vec z]\le \prod_{i=1}^\ell \mathrm{Pr}[M_i\vec{\xi}^{\,(1)}+\dots +M_i\vec{\xi}^{\,(\ell)}=0]^{1/\ell}. $$

In [Reference Ferber, Jain and ZhaoFJZ22], this is proved using Fourier analysis and Hölder’s inequality (the anonymous referee also pointed out that, when $\ell$ is a power of two, this can be proved by iterating the Cauchy–Schwarz inequality). Then, it is not hard to see that

$$ \mathrm{Pr}[M_i\vec{\xi}^{\,(1)}+\dots +M_i\vec{\xi}^{\,(\ell)}=0]=\mathrm{Pr}[\vec{\xi}^{\,(1)}+\dots +\vec{\xi}^{\,(\ell)}=0]=\bigg(2^{-\ell}\binom \ell{\ell/2}\bigg)^k , $$

for each $i=1,\ldots,\ell$ . If we take $\ell=t$ or $\ell=t-1$ (depending on whether t is even or odd), it is a straightforward matter to check that this expression is always at most $t^{-k/2}$ .

Corollary 3.4. Let $0\le d\le k\le n$ and $t\ge 1$ be integers. Consider vectors $\vec{a}_{1},\ldots,\vec{a}_{n}\in\mathbb{R}^{k}$ such that one can form t disjoint bases of $\mathbb{R}^k$ from the vectors $\vec{a}_{1},\ldots, \vec{a}_{n}$ , and let $\mathcal{W}\subseteq \mathbb{R}^k$ be a d-dimensional affine-linear subspace. Letting $\vec{\xi}=(\xi_{1},\ldots,\xi_{n})\in\{-1,1\}^{n}$ be a sequence of independent Rademacher random variables, we then have

$$\mathrm{Pr}[\xi_{1}\vec{a}_{1}+\dots+\xi_{n}\vec{a}_{n}\in\mathcal{W}]\le t^{-(k-d)/2}. $$

Proof. Write $\tilde{\mathcal{W}}$ for the d-dimensional linear subspace parallel to the affine-linear subspace $\mathcal{W}$ (i.e., $\mathcal{W}=\tilde{\mathcal{W}}+\vec{v}$ for some $\vec{v}\in\mathbb{R}^{k}$ ), and consider a linear map $\phi:\mathbb{R}^{k}\to\mathbb{R}^{k-d}$ with kernel $\tilde{\mathcal W}$ (so $\phi(\mathcal{W})$ consists of a single point $\vec{p}\in \mathbb{R}^{k-d}$ ). Writing $M\in\mathbb{R}^{(k-d)\times n}$ for the matrix whose columns are the vectors $\phi(\vec{a}_{1}),\ldots,\phi(\vec{a}_{n})$ , we have

$$\mathrm{Pr}[\xi_{1}\vec{a}_{1}+\dots+\xi_{n}\vec{a}_{n}\in\mathcal{W}]=\mathrm{Pr}[\xi_{1}\phi(\vec{a}_{1})+\dots+\xi_{n}\phi(\vec{a}_{n})=\vec{p}]=\mathrm{Pr}[M\vec{\xi}=\vec{p}]\le t^{-(k-d)/2}, $$

by Theorem 3.3 (note that, for each of the t disjoint bases formed among the vectors $\vec{a}_{1},\ldots, \vec{a}_{n}$ , the image under $\phi$ is a spanning set of $\mathbb{R}^{k-d}$ and hence contains a basis of $\mathbb{R}^{k-d}$ , so we can find t disjoint bases among the columns of M and consequently M contains t disjoint nonsingular $(k-d)\times (k-d)$ submatrices).

Corollary 3.5. Let $1\le k\le n$ be integers and $s \gt 0$ be a real number. Consider a matrix $M\in \mathcal{H}^{k\times n}(s)$ and a vector $\vec{w}\in \mathbb{R}^k$ . Letting $\vec{\xi}=(\xi_{1},\ldots,\xi_{n})\in\{-1,1\}^n$ be a sequence of independent Rademacher random variables, we then have

$$ \mathbb{P}[M\vec{\xi}=\vec{w}]\le (s/k)^{-k/2}. $$

Next, we recall some basic facts about linear forms. A linear form $g\in \mathbb{R}[x_1,\ldots,x_n]$ is a linear polynomial with constant term zero. That is, we can write $g(\vec{x})=\vec{v}\cdot \vec{x}=\vec{v}^{{\intercal}}\vec{x}=\vec{x}^{{\intercal}}\vec{v}$ , where $\vec{v}\in\mathbb{R}^n$ is the coefficient vector of g. Note that, for two linear forms $g_1,g_2\in \mathbb{R}[x_1,\ldots,x_n]$ with coefficient vectors $\vec v_1,\vec v_2\in \mathbb{R}^n$ , we have

(3.1) \begin{equation}g_1(\vec{x})\cdot g_2(\vec{x})=\tfrac{1}{2}g_1(\vec{x})\cdot g_2(\vec{x})+\tfrac{1}{2}g_2(\vec{x})\cdot g_1(\vec{x})=\tfrac{1}{2}\vec{x}^{{\intercal}}\vec{v}_1\vec{v}_2^{{\intercal}}\vec{x}+\tfrac{1}{2}\vec{x}^{{\intercal}}\vec{v}_2\vec{v}_1^{{\intercal}}\vec{x}=\vec{x}^{{\intercal}}A\vec{x},\end{equation}

where $A\in \mathbb{R}^{n\times n}$ is the symmetric matrix given by $A=\frac{1}{2}(\vec{v}_1\vec{v}_2^{{\intercal}}+\vec{v}_2\vec{v}_1^{{\intercal}})$ . More generally, every quadratic form (i.e., every quadratic polynomial with no linear and no constant term), can be written in the form $\vec{x}^{{\intercal}}A\vec{x}$ for a unique symmetric matrix $A\in \mathbb{R}^{n\times n}$ .

Another important ingredient for our proof is the following decoupling lemma.

Lemma 3.6 is a slight variant of a lemma of Costello, Tao and Vu [Reference Costello, Tao and VuCTV06, Lemma 4.7], who popularised decoupling as a tool for polynomial anticoncentration. The particular statement of Lemma 3.6 appears (for example) as [Reference CostelloCos13, Lemma 14]. For the reader’s convenience, we include the proof (which is a simple application of the Cauchy–Schwarz inequality).

Proof of Lemma 3.6. By the Cauchy–Schwarz inequality, we have

\begin{align*} \mathrm{Pr}[\mathcal E(X,Y) \text{ and } \mathcal E(X',Y)]&=\mathbb{E}_Y[\mathrm{Pr}[\mathcal E(X,Y) \text{ and } \mathcal E(X',Y)\,|\, Y]]=\mathbb{E}_Y[\mathrm{Pr}[\mathcal E(X,Y)\,|\, Y]^2]\\ &\ge \mathbb{E}_Y[\mathrm{Pr}[\mathcal E(X,Y)\,|\, Y]]^2=\mathrm{Pr}[\mathcal E(X,Y)]^2. \end{align*}

Taking square roots on both sides gives the desired inequality.

We will also need a simple lemma usually attributed to Odlyzko [Reference OdlyzkoOdl88].

Finally, we will need a simple numerical inequality.

4. Inductive decoupling from a geometric point of view

Note that, for a quadratic polynomial Q and a sequence $\vec{\xi}=(\xi_{1},\ldots,\xi_{n})\in\{-1,1\}^n$ of independent Rademacher random variables, the event $Q(\vec \xi)=0$ is precisely the event that $\vec \xi$ falls in the vanishing locus of Q. Moreover, if the quadratic part of Q has rank less than r (i.e., if Q can be expressed as a linear combination of $r-1$ squares of linear forms, plus an additional linear form, plus a constant term), then it is possible to interpret the event $Q(\vec \xi)=0$ as the event that $\xi_1\vec a_1+\dots +\xi_n\vec a_n\in \mathcal Z$ , where $\vec a_1,\ldots,\vec a_n\in \mathbb{R}^{r}$ are vectors in r-dimensional space, and $\mathcal Z$ is the vanishing locus of some r-variable quadratic polynomial (indeed, the entries of each $\vec a_i$ correspond to the coefficients of $\xi_i$ in each of the linear forms described above).

In this section, we obtain an essentially optimal bound (in Theorem 4.2 below) on probabilities of the form $\mathrm{Pr}[\xi_1\vec a_1+\dots +\xi_n\vec a_n\in \mathcal Z]$ , where $\vec a_1,\ldots,\vec a_n\in \mathbb{R}^{r}$ are vectors satisfying some robust nondegeneracy condition and $\mathcal Z$ is a quadric in $\mathbb{R}^r$ (or more generally, a quadric inside some affine-linear subspace of $\mathbb{R}^r$ ), under the assumption that the dimension r does not grow with n. This may be viewed as a warm-up to the full proof of Theorem 1.1: it is proved via the same inductive decoupling scheme, but has fewer technicalities.

Using the connection described in the first paragraph above (with an additional ‘dropping to a subspace’ argument, of the type we will later see in § 6), one can use Theorem 4.2 to prove Theorem 1.1 in the special case where the quadratic part of Q has bounded rank. Actually, Theorem 4.2 is also an ingredient in the full proof of our main theorem (Theorem 1.1). We also remark that the result of this section fits nicely in the ‘geometric Littlewood–Offord’ framework of Fox, Kwan and Spink [Reference Fox, Kwan and SpinkFKS23]. In particular, the case of Theorem 4.2 where $\mathcal Z\subseteq \mathbb{R}^4$ is a sphere in four dimensions was explicitly raised as the simplest open case of the main problem in [Reference Fox, Kwan and SpinkFKS23].

Before stating the main result of this section, we record some relevant terminology.

For a d-dimensional affine-linear subspace $\mathcal{W}\subseteq \mathbb{R}^{r}$ , we say that $\mathcal{Z}\subsetneq\mathcal{W}$ is a quadric on $\mathcal{W}$ if it is the image of some quadric $\mathrm{V}(P)\subsetneq\mathbb{R}^{d}$ under an affine-linear isomorphism $\phi:\mathbb{R}^{d}\to\mathcal{W}$ . Equivalently, $\mathcal{Z}\subsetneq\mathcal{W}$ is a quadric on $\mathcal{W}$ if and only if $\mathcal{Z}=\mathcal{W}\cap \mathrm{V}(P)$ for some quadratic polynomial $P\in \mathbb{R}[x_1,\ldots,x_r]$ with $\mathcal{W}\not\subseteq \mathrm{V}(P)$ . We say that a quadric $\mathcal{Z}\subsetneq\mathcal{W}$ is irreducible if it is the image of some irreducible quadric $\mathrm{V}(P)\subsetneq\mathbb{R}^{d}$ under an affine-linear isomorphism $\phi:\mathbb{R}^{d}\to\mathcal{W}$ .

As an example, see Figure 1 showing a quadric on a two-dimensional affine plane in $\mathbb{R}^{3}$ . Now we are ready to state the main result of this section.

Figure 1. An example of a quadric on a two-dimensional affine plane in $\mathbb R^3$ .

Theorem 4.2. Let $0\le d<r$ be integers. Let $\mathcal{Z}\subsetneq\mathcal{W}$ be a quadric on a $(d+1)$ -dimensional affine-linear subspace $\mathcal{W}\subseteq\mathbb{R}^{r}$ . Consider vectors $\vec{a}_{1},\ldots,\vec{a}_{n}\in\mathbb{R}^{r}$ such that one can form t disjoint bases of $\mathbb{R}^{r}$ from the vectors $\vec{a}_{1},\ldots, \vec{a}_n$ , where t is a positive integer divisible by $2^d$ . Let $(\xi_1,\ldots,\xi_n)\in\{-1,1\}^{n}$ be a sequence of independent Rademacher random variables. Then

$$\mathrm{Pr}[\xi_{1}\vec{a}_{1}+\dots+\xi_{n}\vec{a}_{n}\in\mathcal{Z}]\le\frac{2^{dr+1}}{t^{(r-d)/2}}. $$

If $\mathcal{Z}\subsetneq\mathcal{W}$ is a quadric on a $(d+1)$ -dimensional affine-linear subspace $\mathcal{W}$ , then $\mathcal{Z}$ has dimension at most d (we do not formally define what ‘dimension’ means in this context, as this is not needed for our arguments, but appeal to the reader’s intuition). Note that the form of the bound in the above theorem (with $t^{(r-d)/2}$ in the denominator) is the same as in Corollary 3.4 for d-dimensional affine-linear subspaces.

For the proof of Theorem 4.2, we will rely on the following algebraic fact.

1. (i) there is a direction $\vec{v}\in \mathbb{R}^d\setminus\{\vec 0\}$ such that $\mathcal{Z}+\mathbb{R}\vec{v}=\mathcal{Z}$ (i.e., $\mathcal{Z}$ is invariant under translation along the direction $\vec{v}$ ); or

2. (ii) for any vectors $\vec{x},\vec{y}\in \mathbb{R}^d$ with $\vec{x}\ne\vec{y}$ , the intersection $(\mathcal{Z}-\vec{x})\cap(\mathcal{Z}-\vec{y})$ is a quadric on a $(d-1)$ -dimensional affine-linear subspace $\mathcal{W}_{\vec{x},\vec{y}}\subsetneq\mathbb{R}^{d}$ .

Proof. Let $\mathcal{Z}=\mathrm{V}(P)$ for a nonzero quadratic polynomial $P\in \mathbb{R}[x_1,\ldots,x_d]$ , and suppose that (i) does not hold. Note that then $\mathcal{Z}-\vec x=\{\vec{w}\in \mathbb{R}^d: P(\vec w+\vec x)=0\}$ for any $\vec x\in \mathbb{R}^d$ . For any $\vec{x},\vec{y}\in \mathbb{R}^d$ with $\vec{x}\ne\vec{y}$ , consider the linear polynomial $L_{\vec{x},\vec{y}}:\vec w\mapsto P(\vec{w}+\vec{y})-P(\vec{w}+\vec{x})$ and write $\mathcal{W}_{\vec{x},\vec{y}}=\mathrm{V}(L_{\vec{x},\vec{y}})$ , so $(\mathcal{Z}-\vec{x})\cap(\mathcal{Z}-\vec{y})=(\mathcal{Z}-\vec{x})\cap\mathcal{W}_{\vec{x},\vec{y}}$ . This already shows that $(\mathcal{Z}-\vec{x})\cap(\mathcal{Z}-\vec{y})$ is the vanishing locus of a quadratic polynomial on $\mathcal{W}_{\vec{x},\vec{y}}$ . To verify (ii), we will check that $L_{\vec{x},\vec{y}}$ is not the zero polynomial (which shows that $\dim \mathcal{W}_{\vec{x},\vec{y}}=d-1$ ), and that $\mathcal{W}_{\vec{x},\vec{y}}\not \subseteq \mathcal{Z}-\vec x$ (which implies $(\mathcal{Z}-\vec{x})\cap(\mathcal{Z}-\vec{y})=(\mathcal{Z}-\vec{x})\cap\mathcal{W}_{\vec{x},\vec{y}}\subsetneq \mathcal{W}_{\vec{x},\vec{y}}$ ).

First, the reason $L_{\vec{x},\vec{y}}$ cannot be the zero polynomial is that (i) does not hold. Indeed, if $L_{\vec{x},\vec{y}}$ were the zero polynomial, then for all $\vec{w}\in\mathbb{R}^{d}$ and $\lambda\in\mathbb{Z}$ we would have $P(\vec{w})=P(\vec{w}+\lambda(\vec{x}-\vec{y}))$ . That is to say, for all $\vec{w}\in\mathbb{R}^{d}$ the quadratic polynomial $\lambda\mapsto P(\vec{w})-P(\vec{w}+\lambda(\vec{x}-\vec{y}))$ would have infinitely many zeroes, so would be the zero polynomial, meaning that $P(\vec{w})=P(\vec{w}+\lambda(\vec{x}-\vec{y}))$ for all $\lambda\in\mathbb{R}$ and $\vec{w}\in\mathbb{R}^{d}$ . Hence we would have $\mathcal{Z}+\lambda(\vec{x}-\vec{y})=\mathcal{Z}$ for all $\lambda\in\mathbb{R}$ , so (i) would hold for $\vec{v}=\vec{x}-\vec{y}$ .

Finally, it remains to show $\mathcal{W}_{\vec{x},\vec{y}}\not \subseteq \mathcal{Z}-\vec x$ . Indeed, otherwise $\mathcal{W}_{\vec{x},\vec{y}}$ would be an irreducible component of $\mathcal{Z}-\vec x$ , but by our assumptions $\mathcal{Z}-\vec x$ is irreducible (since $\mathcal{Z}$ is). We also have $\mathcal{Z}-\vec x\ne \mathcal{W}_{\vec{x},\vec{y}}$ , since $\mathcal{Z}-\vec x$ not invariant under translation along the direction of any nonzero vector (since $\mathcal{Z}$ does not satisfy (i)). So we indeed have $\mathcal{W}_{\vec{x},\vec{y}}\not \subseteq \mathcal{Z}-\vec x$ .

Now we prove Theorem 4.2.

Step 1: the reducible case. First, it is easy to handle the case where the quadric $\mathcal{Z}\subsetneq \mathcal{W}$ is reducible, i.e., where the irreducible components of $\mathcal{Z}$ are two affine-linear subspaces of dimension d. Indeed, suppose that $\mathcal{Z}=\mathcal{V}_{1}\cup\mathcal{V}_{2}$ for two d-dimensional affine-linear subspaces $\mathcal{V}_{1},\mathcal{V}_{2}\subsetneq\mathcal{W}$ . By Corollary 3.4 (i.e., by the version of Halász’ theorem in [Reference Ferber, Jain and ZhaoFJZ22]), we have

$$\mathrm{Pr}[\vec{X}\in\mathcal{Z}]\le\mathrm{Pr}[\vec{X}\in\mathcal{V}_{1}]+\mathrm{Pr}[\vec{X}\in\mathcal{V}_{2}]\le \frac{2}{t^{(r-d)/2}}, $$

which implies the desired result.

Step 2: the translation-invariant case. It is also easy to handle the case where there is a direction $\vec{v}\ne\vec 0$ such that $\mathcal{Z}+\mathbb{R}\vec{v}=\mathcal{Z}$ , because then we can project our entire problem along the direction of $\vec{v}$ to obtain a lower-dimensional problem (noting that then we also have $\mathcal{W}+\mathbb{R}\vec{v}=\mathcal{W}$ ). Indeed, consider a linear map $\phi:\mathbb{R}^r\to \mathbb{R}^{r-1}$ with kernel $\operatorname{span}(\vec v)$ , and observe that $\phi(\mathcal{Z})$ is a quadric on the $(d-1)$ -dimensional affine-linear subspace $\phi(\mathcal{W})$ . Then, we have

$$\mathrm{Pr}[\vec{X}\in\mathcal{Z}]=\mathrm{Pr}[\phi(\vec{X})\in\phi(\mathcal{Z})]\le\frac{2^{(d-1)(r-1)+1}}{t^{(r-d)/2}}, $$

by our induction hypothesis (noting that among $\phi(\vec{a}_1),\ldots, \phi(\vec{a}_n)$ one can still form t disjoint bases), and the desired result follows.

Step 3: decoupling. Now, we can assume that $\mathcal{Z}\subseteq \mathcal{W}$ is an irreducible quadric and that there is no direction $\vec{v}\ne\vec 0$ such that $\mathcal{Z}+\mathbb{R}\vec{v}=\mathcal{Z}$ . Let $\tilde{\mathcal{W}}\subseteq \mathbb{R}^r$ be the $(d+1)$ -dimensional linear subspace parallel to $\mathcal{W}$ (i.e., $\mathcal{W}=\tilde{\mathcal{W}}+\vec{w}$ for some $\vec{w}\in\mathbb{R}^{r}$ ). For any $\vec{x}\ne\vec{y}$ with $\vec x-\vec y\in \tilde{\mathcal W}$ , by Lemma 4.3 (with an affine-linear isomorphism $\mathbb{R}^{d}\to \mathcal W-\vec x$ ), the intersection $(\mathcal{Z}-\vec{x})\cap(\mathcal{Z}-\vec{y})$ is a quadric on a d-dimensional affine-linear subspace $\mathcal{W}_{\vec{x},\vec{y}}\subsetneq \tilde{\mathcal{W}}+\vec w_{\vec{x},\vec{y}}$ for some $\vec w_{\vec{x},\vec{y}}\in \mathbb{R}^r$ . The next step is to split our random variable into two parts and use the decoupling lemma (Lemma 3.6) to relate our probability $\mathrm{Pr}[\vec{X}\in\mathcal{Z}]$ to probabilities that certain random variables lie in quadrics of the form $(\mathcal{Z}-\vec{x})\cap(\mathcal{Z}-\vec{y})$ (these probabilities can then be bounded via the induction hypothesis).

Let $[n]=I\cup J$ be a partition of the index set [n] into two subsets I,J such that one can form $t/2$ disjoint bases from the vectors $\vec a_i$ for $i\in I$ , and one can also form $t/2$ disjoint bases from the vectors $\vec a_j$ for $j\in J$ . Let $\vec{X}_{I}=\sum_{i\in I}\xi_{i}\vec{a}_{i}$ and let $\vec{X}_{J}=\sum_{j\in J}\xi_{j}\vec{a}_{j}$ (so $\vec{X}=\vec{X}_I+\vec{X}_J$ ) and let $\vec{X}_I'$ be an independent copy of the random variable $\vec{X}_I$ . By decoupling (Lemma 3.6) we have

(4.1) \begin{align}\mathrm{Pr}[\vec{X}\in\mathcal{Z}]^{2} & \le\mathrm{Pr}[\vec{X}_I+\vec{X}_J\in\mathcal{Z} \text{ and } \vec{X}_I'+\vec{X}_J\in\mathcal{Z}]\nonumber \\ & =\mathrm{Pr}[\vec{X}_J\in(\mathcal{Z}-\vec{X}_I)\cap(\mathcal{Z}-\vec{X}_I') \text{ and } \vec{X}_I\ne\vec{X}_I']+\mathrm{Pr}[\vec{X}_I+\vec{X}_J\in\mathcal{Z} \text{ and } \vec{X}_I'=\vec{X}_I].\end{align}

Step 4: dealing with degenerate intersection. We now study the second term in Equation (4.1) (which can be thought of as a lower-order ‘error term’ corresponding to the possibility that decoupling does not actually reduce the dimension of our problem).

For any outcome of $\vec{X}_I$ , we have $\mathrm{Pr}[\vec{X}_I'=\vec{X}_I\,|\,\vec{X}_I]\le(t/2)^{-r/2}$ by Halász’ theorem (Corollary 3.4, taking $\mathcal{W}$ to be a single point). So,

(4.2) \begin{equation}\mathrm{Pr}[\vec{X}_I+\vec{X}_J\in\mathcal{Z} \text{ and } \vec{X}_I'=\vec{X}_I]\le \mathrm{Pr}[\vec{X}_I+\vec{X}_J\in\mathcal{Z}]\cdot (t/2)^{-r/2}=\mathrm{Pr}[\vec{X}\in\mathcal{Z}]\cdot (t/2)^{-r/2}.\end{equation}

Step 5: inductively bounding the main term. Now we deal with the first term in Equation (4.1). Recalling that $\tilde{\mathcal{W}}\subseteq \mathbb{R}^r$ is the $(d+1)$ -dimensional linear subspace parallel to $\mathcal{W}$ , note that it is impossible to have $\vec{X}_J\in(\mathcal{Z}-\vec{X}_I)\cap(\mathcal{Z}-\vec{X}_I')$ if $\vec{X}_I-\vec{X}_I'\not\in\tilde{\mathcal{W}}$ (since then $(\mathcal{Z}-\vec{X}_I)\cap(\mathcal{Z}-\vec{X}_I')\subseteq (\mathcal{W}-\vec{X}_I)\cap(\mathcal{W}-\vec{X}_I')=\emptyset$ ). So, we combine our induction hypothesis with a bound on the probability that $\vec{X}_I-\vec{X}_I'\in\tilde{\mathcal{W}}$ .

For all outcomes of $\vec{X}_I'$ , by Halász’ theorem (Corollary 3.4) we have

$$ \mathrm{Pr}[\vec{X}_I-\vec{X}_I'\in\tilde{\mathcal{W}}\,|\,\vec{X}_I']=\mathrm{Pr}[\vec{X}_I\in\tilde{\mathcal{W}}+\vec{X}_I'\,|\,\vec{X}_I']\le(t/2)^{-(r-d-1)/2}. $$

Also, for any outcomes of $\vec{X}_I,\vec{X}_I'$ such that $\vec{X}_I\ne \vec{X}_I'$ and $\vec{X}_I-\vec{X}_I'\in\tilde{\mathcal{W}}$ , by the discussion in Step 3, the intersection $(\mathcal{Z}-\vec{X}_I)\cap(\mathcal{Z}-\vec{X}_I')$ is a quadric on a d-dimensional affine-linear subspace $\mathcal{W}_{\vec{X}_I,\vec{X}_I'}\subseteq \mathbb{R}^r$ . So by our induction hypothesis we have

$$ \mathrm{Pr}[\vec{X}_J\in(\mathcal{Z}-\vec{X}_I)\cap(\mathcal{Z}-\vec{X}_I')\,|\,\vec{X}_I,\vec{X}_I']\le\frac{2^{(d-1)r+1}}{(t/2)^{(r-d+1)/2}}. $$

Thus, we obtain

(4.3) \begin{align} \mathrm{Pr}[\vec{X}_J\in(\mathcal{Z}-\vec{X}_I)\cap(\mathcal{Z}-\vec{X}_I') \text{ and } \vec{X}_I\ne\vec{X}_I'] & \le (t/2)^{-(r-d-1)/2}\cdot\frac{2^{(d-1)r+1}}{(t/2)^{(r-d+1)/2}} \nonumber\\ & =\frac{2^{(d-1)r+1}}{(t/2)^{r-d}}\le\frac{2^{dr+1}}{t^{r-d}}.\end{align}

Step 6: concluding. We can now deduce from (4.1), (4.2) and (4.3) that

$$ \mathrm{Pr}[X\in\mathcal{Z}]^{2} \le\mathrm{Pr}[X\in\mathcal{Z}]\cdot(t/2)^{-r/2}+\frac{2^{dr+1}}{t^{r-d}}=\mathrm{Pr}[X\in\mathcal{Z}]\cdot \frac{2^{r/2}}{t^{r/2}}+\frac{2^{dr+1}}{t^{r-d}}. $$

So, by Lemma 3.8 we have (also using that $d\ge 1$ and $r\ge d+1\ge 2$ )

$$\mathrm{Pr}[X\in\mathcal{Z}]\le \frac{2^{r/2}}{t^{r/2}}+\frac{2^{(dr+1)/2}}{t^{(r-d)/2}}\le\frac{2^{dr+1}}{t^{(r-d)/2}}, $$

as desired.

5. Proof strategy for the general case

Now we turn to the general (not necessarily low-rank) case of Theorem 1.1. Actually, we consider the following variation on Theorem 1.1, with a slightly more technical assumption on Q (namely, we need to assume that the quadratic part of Q ‘robustly depends on many different variables’). This assumption is very similar to assumptions in some previous Littlewood–Offord-type theorems [Reference Meka, Nguyen and VuMNV16, Reference Razborov and ViolaRV13].

Theorem 5.1. Let $Q\in\mathbb{R}[x_{1},\ldots,x_{n}]$ be a multivariate quadratic polynomial with quadratic part $\vec x^{\intercal} A\vec x$ for some symmetric matrix $A\in \mathbb{R}^{n\times n}$ . Let $s\ge 1$ be an integer, and assume that, for every subset $S\subseteq[n]$ with $|S|\ge n-s$ , the submatrix $A[S\times S]$ has at least one nonzero entry outside its diagonal. Then, for a sequence $\vec{\xi}\in\{-1,1\}^{n}$ of independent Rademacher random variables, we have

$$\mathrm{Pr}[Q(\vec{\xi})=0]\le\frac{C'}{\sqrt{s}}, $$

for some absolute constant C’.

The reason that the assumption in Theorem 5.1 only pays attention to the nondiagonal entries of A is that the diagonal entries correspond to square terms of the form $x_i^2$ . If $x_i\in \{-1,1\}$ then $x_i^2$ is always equal to 1, so such square terms can be treated as constants (and therefore they ‘do not really contribute’ to the quadratic part of Q).

In § 11 we will show how to deduce Theorem 1.1 (and Theorem 1.2 for general distributions) from Theorem 5.1. For most of the rest of the paper, we will focus on proving Theorem 5.1.

At a high level, the strategy to prove Theorem 5.1 is similar to the proof of Theorem 4.2: in order to estimate the probability $\mathrm{Pr}[Q(\vec{\xi})=0]$ for a given quadratic polynomial $Q\in \mathbb{R}[x_1,\ldots,x_n]$ , we inductively estimate probabilities of the form $\mathrm{Pr}[Q(\vec{\xi})=0\text{ and }M\vec{\xi}=\vec{w}]$ for a given matrix $M\in \mathbb{R}^{k\times n}$ and a given vector $\vec{w}\in \mathbb{R}^k$ (i.e., we estimate the probability that $\vec{\xi}$ lies in a given quadric on a given affine-linear subspace). However, there are additional difficulties in comparison with the proof of Theorem 4.2 in the previous section.

Most importantly, recall that in Theorem 4.2 we worked with a random vector $\xi_1\vec a_1+\cdots \xi_n\vec a_n$ which has ‘a lot of anticoncentration in each direction’ (we assumed that there are many disjoint bases among the vectors $\vec a_1,\ldots,\vec a_n$ ). This was only possible since the dimension of our space was much less than n: to be precise, that proof approach can only obtain bounds of the form $O(1/\sqrt n)$ when we have $\Omega(n)$ disjoint bases among the vectors $\vec a_1,\ldots,\vec a_n$ , which is only possible when we are working in O(1)-dimensional space.

In the proof of Theorem 5.1 we work directly with the random vector $\vec \xi\in \{-1,1\}^n$ in n-dimensional space. We cannot ensure good anticoncentration in all directions, so we need to restrict the affine-linear subspaces we can consider. Specifically, we prove bounds on $\mathrm{Pr}[Q(\vec{\xi})=0$ and $M\vec{\xi}=\vec{w}]$ only when the matrix M satisfies a Halász-type robust rank condition (as in Definition 3.1). This means that we now need to maintain this robust rank condition as M changes over the course of the induction.

Also, we encounter much more delicate nondegeneracy issues than in the proof of Theorem 4.2. In particular, even if Q satisfies the nondegeneracy condition in Theorem 5.1 (robustly depending on many variables), it may become very degenerate when restricted to a subspace of the form $\{\vec x :M\vec x=\vec w\}$ for a matrix $M\in \mathbb{R}^{k\times n}$ and a vector $\vec{w}\in \mathbb{R}^k$ . For example, consider the case where n is divisible by 2 and $Q\in \mathbb R[x_1,\ldots,x_n]$ , $M\in \mathbb R^{1\times n}$ and $\vec w\in\mathbb R^1$ are defined by

(5.1) \begin{equation}Q(\vec x)=(x_1+\dots+x_{n/2})(x_{n/2+1}+\dots+x_{n}),\quad\quad M=(1,\ldots,1,0,\ldots,0),\quad\quad \vec w=0;\end{equation}

(where the first $n/2$ entries of $M\in \mathbb R^{1\times n}$ are ‘1’ and the last $n/2$ entries are ‘0’). In this case Q is always zero on the subspace $\{\vec x\in \mathbb R^n:M\vec x=\vec w\}$ . So, in order to be able to obtain a sensible bound, we need to ensure that Q satisfies a nondegeneracy condition with respect to M.

In the rest of this section we describe the strategy of the proof of Theorem 5.1 in a bit more detail, stating several key lemmas and definitions along the way. First, we elaborate on the ‘nondegeneracy with respect to M’ condition mentioned above. To formulate this condition (as well as other similar conditions appearing later in the proof), we define the notion of a (T,U)-perturbation, as follows.

As the degenerate $k=0$ case of the above definition, note that if $T\in \mathbb R^{0\times m}$ and $U\in \mathbb R^{0\times n}$ are ‘empty matrices’, then $A'\in \mathbb{R}^{n\times m}$ is a (T,U)-perturbation of A if and only if $A'=A$ .

For $Q,M,\vec w$ as defined in the example in Equation (5.1), we can write $Q(\vec x)=\vec x^{\intercal} A \vec x$ for a symmetric matrix $A\in \mathbb{R}^{n\times n}$ (with a block structure where the bottom left block and the top right block, each of size $(n/2)\times (n/2)$ , have all entries being $1/2$ , and all entries outside these blocks are 0). It turns out that this matrix A is an (M,M)-perturbation of the zero matrix in $\mathbb{R}^{n\times n}$ . Roughly speaking, this is the reason for the degenerate behaviour of Q on the subspace $\{\vec x :M\vec x=\vec w\}$ . In general, the notion of an (M,M)-perturbation gives a condition under which two $n\times n$ matrices give rise to quadratic polynomials which are ‘essentially the same’ on an affine-linear subspace of the form $\{\vec x :M\vec x=\vec w\}$ , as follows.

Consider an (M,M)-perturbation A’ of A, and let $A^*\in \mathbb{R}^{n\times n}$ be a matrix which agrees with A’ on its off-diagonal entries. Then, there is a quadratic polynomial $Q^*\in [x_1,\ldots,x_n]$ with quadratic part $\vec x^{\intercal} A^*\vec x$ such that $Q^*(\vec{\xi})=Q(\vec{\xi})$ for all $\vec{\xi}\in \{-1,1\}^n$ with $M\vec{\xi}=\vec{w}$ . In particular, for a sequence $\vec{\xi}\in\{-1,1\}^{n}$ of independent Rademacher random variables, we have

$$ \mathrm{Pr}[Q(\vec{\xi})=0 \text{ and } M\vec{\xi}=\vec{w}]=\mathrm{Pr}[Q^*(\vec{\xi})=0 \text{ and } M\vec{\xi}=\vec{w}]. $$

Proof. Let $A^*=A'+D=A+LM+M^{{\intercal}}R+D$ for some matrices $L\in\mathbb{R}^{n\times k}$ and $R\in\mathbb{R}^{k\times n}$ and a diagonal matrix $D\in \mathbb{R}^{n\times n}$ , and let $\lambda_1,\ldots,\lambda_n\in \mathbb{R}$ be the diagonal entries of D. Then

\begin{align*}\vec x^{\intercal} A^*\vec x & =\vec x^{\intercal} A\vec x+\vec x^{\intercal} LM\vec x+\vec x^{\intercal} M^{{\intercal}}R\vec x+\vec x^{\intercal} D\vec x \nonumber\\& =\vec x^{\intercal} A\vec x+(M\vec x)^{\intercal} L^{\intercal} \vec x+(M\vec x)^{\intercal} R \vec x +(\lambda_1x_1^2+\dots+\lambda_nx_n^2).\end{align*}

Thus, for every $\vec{\xi}\in \{-1,1\}^n$ with $M\vec{\xi}=\vec{w}$ , we have

$$ \vec \xi^{\intercal} A^*\vec \xi=\vec \xi^{\intercal} A\vec \xi+\vec w^{\intercal} L^{\intercal} \vec \xi+\vec w^{\intercal} R \vec x+(\lambda_1\xi_1^2+\dots+\lambda_n\xi_n^2)=\vec \xi^{\intercal} A\vec \xi+\vec w^{\intercal} (L^{\intercal}+R) \vec \xi+(\lambda_1+\dots+\lambda_n). $$

Hence, we can define the desired quadratic polynomial $Q^*\in [x_1,\ldots,x_n]$ with quadratic part $\vec x^{\intercal} A^*\vec x$ by

$$ Q^*(\vec x)=Q(\vec x)+\vec x^{\intercal} A^*\vec x-\vec x^{\intercal} A\vec x-\vec w^{\intercal} (L^{\intercal}+R) \vec x-(\lambda_1+\dots+\lambda_n), $$

and we indeed have $Q^*(\vec{\xi})=Q(\vec{\xi})$ for all $\vec{\xi}\in \{-1,1\}^n$ with $M\vec{\xi}=\vec{w}$ .

Now, recall that our strategy to prove Theorem 5.1 is to inductively upper-bound probabilities of the form $\mathrm{Pr}[Q(\vec{\xi})=0\text{ and }M\vec{\xi}=\vec w]$ , assuming that M satisfies a robust rank condition, and assuming a nondegeneracy condition on Q with respect to M. It will be convenient to introduce some notation for the maximum possible such probability, as follows.

Definition 5.4. For an integer $k\ge 0$ and real number $s\ge 0$ , let us define

$$ f(k,s)=\sup_{(n,Q,M,\vec w)} \mathrm{Pr}[Q(\vec{\xi})=0 \text{ and } M\vec{\xi}=\vec w], $$

where the supremum is taken over all quadruples $(n,Q,M,\vec w)$ , where n is a positive integer, $Q\in\mathbb{R}[x_{1},\ldots,x_{n}]$ is a quadratic polynomial, $M\in\mathcal{H}^{k\times n}(s)$ and $\vec w\in \mathbb{R}^k$ , such that the following condition holds:

(*) If we write the quadratic part of $Q(\vec{x})$ as $\vec{x}^{{\intercal}}A\vec{x}$ for a symmetric matrix $A\in \mathbb{R}^{n\times n}$ , then for every subset $S\subseteq[n]$ with $|S|\ge n-s$ , and every (M,M)-perturbation A’ of A, the submatrix $A'[S\times S]$ has at least one nonzero entry outside the diagonal.

For each such quadruple $(n,Q,M,\vec w)$ , the probability above is taken with respect to a sequence of independent Rademacher random variables $\vec{\xi}\in\{-1,1\}^{n}$ .

Note that we always have $f(k,s)\le 1$ , since f(k,s) is defined as a supremum of certain probabilities (which are all at most 1). Also note that for $0\le s'\le s$ , we always have $f(k,s)\le f(k,s')$ (since the supremum in the definition of f(k,s’) is taken over a wider range of quadruples $(n,Q,M,\vec w)$ than for f(k,s)).

Note that for a polynomial $Q\in \mathbb{R}[x_1,\ldots,x_n]$ and an integer $s\ge 1$ as in Theorem 5.1, condition ( $*$ ) is satisfied for $k=0$ , the empty matrix $M\in \mathbb{R}^{0\times n}$ and the empty vector $\vec{w}\in \mathbb{R}^0$ . Indeed, writing the quadratic part of Q as $\vec{x}^{{\intercal}}A\vec{x}$ for a symmetric matrix $A\in \mathbb{R}^{n\times n}$ , the only (M,M) perturbation A’ of A is $A'=A$ , and by the assumption in Theorem 5.1 the matrix $A'[S\times S]=A[S\times S]$ has at least one nonzero entry outside the diagonal for every subset $S\subseteq[n]$ with $|S|\ge n-s$ . Therefore, we have (noting that the condition $M\vec \xi=\vec w\in \mathbb{R}^0$ is vacuous)

$$ \mathrm{Pr}[Q(\vec \xi)=0]=\mathrm{Pr}[Q(\vec{\xi})=0 \text{ and } M\vec{\xi}=\vec w]\le f(0,s). $$

Thus, proving Theorem 5.1 amounts to showing that $f(0,s)\le C'/\sqrt{s}$ for some absolute constant C’.

Now, the following recursive upper bound on f(k,s) is the main ingredient in our proof of Theorem 5.1.

Theorem 5.5. For any integer $k\ge 0$ and any real number $s \gt 0$ , we have

$$ f(k,s)\le \max\{s_*^{-(k+1)/2}, s_*^{-(k+2)/2}+s_*^{-k/4}\cdot f(k+1,s_*)^{1/2}\}, $$

where $s_*=s/(k+2)^{500}$ .

Using this recursive bound, we can obtain an upper bound for f(k,s) inductively (the formula for this upper bound is rather complicated, so we postpone this calculation to § 10). With another straightforward (although somewhat technical) calculation, one can then show that in the $k=0$ case this upper bound implies Theorem 5.1. This latter calculation can also be found in § 10.

To prove Theorem 5.5, we need to upper-bound $\mathrm{Pr}[Q(\vec{\xi})=0\text{ and }M\vec{\xi}=\vec w]$ for $Q,M,\vec w$ satisfying the conditions in Definition 5.4, in terms of a probability of the same form but with slightly different parameters (most importantly, with ‘ $k+1$ ’ in place of ‘k’). To do so, we use our inductive decoupling scheme outlined in § 2.3: we consider a partition $[n]=I\cup J$ of the index set into two parts I and J, and, using the decoupling inequality in Lemma 3.6, we will obtain an upper bound on $\mathrm{Pr}[Q(\vec{\xi})=0\text{ and }M\vec{\xi}=\vec w]$ involving conditional probabilities of the form

(5.2) \begin{equation}\mathrm{Pr}[Q_{\vec \xi[I]}(\vec{\xi}[J])=0 \text{ and } M_{\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}\vec{\xi}[J]=\vec w_{\xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}\,|\,\xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]],\end{equation}

where $M_{\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}\in \mathbb{R}^{(k+1)\times J}$ is a $(k+1)\times |J|$ matrix depending on $\vec \xi[I]$ and $\vec{\xi}\mkern2mu\vphantom{\xi}'[I]$ , and $Q_{\vec \xi[I]}\in \mathbb{R}[x_j: j\in J]$ is a quadratic polynomial whose coefficients depend on $\vec \xi[I]$ , and $\vec w_{\xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}\in \mathbb{R}^{k+1}$ is a vector depending on $\vec \xi[I]$ and $\vec{\xi}\mkern2mu\vphantom{\xi}'[I]$ . In our proof of Theorem 5.5, we wish to upper-bound conditional probabilities of the form in Equation (5.2) by $f(k+1,s_*)$ . Recalling Definition 5.4, this requires that $M_{\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}\in \mathcal H^{(k+1)\times n}(s_*)$ (i.e., that $M_{\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}$ robustly has rank at least $k+1$ ). So, as previously outlined, an important ingredient in the proof is a ‘robust rank inheritance’ lemma, which implies that this robust rank condition for $M_{\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}$ holds with sufficiently high probability.

Now, the way in which $M_{\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}$ is derived from M depends on Q: specifically, one can check that $M_{\vec \xi[I],\vec{\xi}\mkern2mu\vphantom{\xi}'[I]}$ is obtained from $M[[k]\times J]$ by adding the vector $2A[J\times I](\vec{\xi}[I]-\vec{\xi}\mkern2mu\vphantom{\xi}'[I])$ as an additional row, where $\vec x^{\intercal} A \vec x$ is the quadratic part of $Q(\vec x)$ . The statement of our robust rank inheritance lemma requires an assumption on Q; to specify this assumption we need another definition.

1. (a) For $t=1,\ldots,s$ , the submatrix $T[[k]\times I_t]$ has rank k.

2. (b) For $t=1,\ldots,s$ , the submatrix $U[[k]\times J_t]$ has rank k.

3. (c) For $t=1,\ldots,s$ , every $(T[[k]\times I_t],U[[k]\times J_t])$ -perturbation of the matrix $A[J_t\times I_t]$ has rank at least r.

In our proof of Theorem 5.5, we consider a partition $[n]=I\cup J$ as outlined above, and take the matrix ‘A’ in Definition 5.6 to be the matrix $A[J\times I]$ together with $T=M[[k]\times I]$ and $U=M[[k]\times J]$ . In this case, $(M[[k]\times I],M[[k]\times J],A[J\times I])\in \mathcal{M}_r^{k,I,J}(s)$ says (roughly speaking) that $A'[J\times I]$ robustly has rank at least r for any (M,M)-perturbation A’ of A.

Now, our robust rank inheritance lemma is as follows.

Lemma 5.7. Let $s\ge 1$ and $0\le k\le m\le n$ be integers. Consider $(T,U,A)\in\mathcal{M}_2^{k,m,n}(s)$ , and vectors $\vec{y}\in\mathbb{R}^{k}$ and $\vec{b}\in\mathbb{R}^{n}$ . Let $\vec{\xi}\in\{-1,1\}^{m}$ be a sequence of independent Rademacher random variables, and write $U_{\vec \xi}'\in \mathbb{R}^{(k+1)\times n}$ for the (random) matrix obtained by appending the vector $A\vec{\xi}-\vec{b}$ as an additional row to U. Then we have

$$\mathrm{Pr}[T\vec{\xi}=\vec{y} \text{ and } U_{\vec \xi}'\notin \mathcal H^{(k+1)\times n}(s/6)]\le \bigg(\frac{s}{10^{61}(k+2)^{20}}\bigg)^{-(k+2)/2}. $$

We will prove Lemma 5.7 in § 7. We remark that on the right-hand side of the inequality, both the ‘ $+2$ ’ in the exponent $(k+2)/2$ and the ‘ $+2$ ’ in the denominator are due to the ‘2’ in the assumption $(T,U,A)\in\mathcal{M}_2^{k,m,n}(s)$ ; in general we would get a ‘ $+r$ ’ in both of these places if we assumed that $(T,U,A)\in\mathcal{M}_r^{k,m,n}(s)$ . The exponent $(k+2)/2$ here leads to the exponent $(k+2)/2$ appearing in Theorem 5.5, and the ‘ $+2$ ’ there is crucial in order to deduce Theorem 5.1 from Theorem 5.5 (just having ‘ $+1$ ’ there would not suffice).

In order to actually apply Lemma 5.7 in the proof of Theorem 5.5, we would like to show that there is a partition $[n]=I \cup J$ such that $(M[[k]\times I],M[[k]\times J],A[J\times I])\in\mathcal{M}_2^{k,I,J}(s)$ for a suitable value of s. Such a partition might not in general exist, because we are making no assumption that A itself even has rank at least two. However, if every (M,M)-perturbation of A robustly has rank at least two even after changing the diagonal entries, we can find a suitable partition $[n]=I \cup J$ using the following lemma.

Lemma 5.8. Let $0\le k\le n$ be integers and let $s\ge 4k+8$ be a real number. Let $M\in\mathcal{H}^{k\times n}(s)$ and let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix such that $\operatorname{rank} A^*[S\times S]\ge 2$ for any subset $S\subseteq [n]$ of size $|S|\ge n- s$ and any matrix $A^*\in \mathbb{R}^{n\times n}$ that agrees with some (M,M)-perturbation of A in all off-diagonal entries. Then there is a partition $[n]= I\cup J$ , with $|I|\le s$ , such that

$$ (M[[k]\times I],M[[k]\times J],A[J\times I])\in\mathcal{M}_2^{k,I,J}(\lfloor s/(4k+8)\rfloor). $$

We prove Lemma 5.8 in § 8. Roughly speaking, we are able to greedily find the desired subsets $I_1,\ldots,I_s,J_1,\ldots,J_s$ in the definition of $\mathcal{M}_2^{k,m,n}(s)$ in Definition 5.6.

To summarise, the proof of Theorem 5.5 combines a number of ingredients. If A does not satisfy the assumption in Lemma 5.8 (i.e., if some (M,M)-perturbation of A does not robustly have rank at least two after changing the diagonal entries), then we complete the proof using Theorem 4.2 (in the low-rank case, there are various types of degeneracies to consider, which can be gracefully handled with a geometric point of view). Otherwise, we apply Lemma 5.8 to find a suitable partition $I\cup J$ , and we apply the decoupling inequality in Lemma 3.6 to $(\vec \xi[I],\vec \xi[J])$ . We manipulate the resulting expression in roughly the same way as in the proof of Theorem 4.2, and then bound relevant quantities using Lemma 5.7 and Corollary 3.5. After proving Theorem 5.5, we can obtain Theorem 5.1 with somewhat technical calculations (presented in § 10).

In the next section we state and prove a consequence of Theorem 4.2 which will be suitable to handle the low-rank case of Theorem 5.5. In § 7 we prove Lemma 5.7 and in § 8 we prove Lemma 5.8. The proof of Theorem 5.5 then appears in § 9.

6. The low-rank case

In this section, we show how to use Theorem 4.2 to deduce the following proposition, which handles the low-rank case of Theorem 5.5.

Then for a sequence $\vec{\xi}\in\{-1,1\}^{n}$ of independent Rademacher random variables we have

(6.1) \begin{equation}\mathrm{Pr}[Q(\vec \xi)=0 \text{ and } M\vec{\xi}=\vec{w}]\le\bigg(\frac{s}{2^{3r^{2}}(k+r)^2}\bigg)^{-(k+1)/2}.\end{equation}

In our proof of Theorem 5.5, we will use Proposition 6.1 only for $r=5$ , but we still state it here for general r (as the proof works for any r).

Deducing Proposition 6.1 from Theorem 4.2 mostly consists of translating from ‘geometric’ to ‘algebraic’ language. To give a some idea of how this works: note that under the assumptions of Proposition 6.1, we can write

$$ Q(\vec x)=\lambda_{1}\cdot (g^{({1})}(\vec{x}))^2+\dots+\lambda_{r-1} \cdot (g^{({r-1})}(\vec{x}))^2+g^{(r)}(\vec x)+c , $$

for some $\lambda_1,\ldots,\lambda_{r-1},c\in \mathbb{R}$ and some linear forms $g^{({1})},\ldots,g^{({r-1})},g^{(r)} \in \mathbb{R}[x_1,\ldots,x_n]$ . That is to say, we can write

$$ Q(\vec x)=P(g^{(1)}(\vec{x}),\ldots,g^{(r)}(\vec{x}))=P(x_1\vec a_1+\dots+x_n\vec a_n), $$

for the quadratic polynomial $P(\vec{y})=\lambda_1 y_1^2+\dots+\lambda_{r-1} y_{r-1}^2+y_r+c$ , where for $j=1,\ldots,n$ , the vector $\vec a_j\in \mathbb R^r$ records the coefficients of $x_j$ in the linear forms $g^{({1})},\ldots,g^{(r)}$ . So, the event $\{Q(\vec \xi)=0\}$ can be interpreted as the event that $\xi_1\vec a_1+\dots+\xi_n\vec a_n$ falls in the vanishing locus of the polynomial P. The joint event $\{Q(\vec \xi)=0\text{ and }M\vec\xi=\vec w \}$ can be given a similar geometric interpretation: roughly speaking, we consider k additional linear forms $g^{({r+1})},\ldots,g^{({r+k})}\in \mathbb{R}[x_1,\ldots,x_n]$ corresponding to the k rows of M, augmenting $\vec a_1,\ldots,\vec a_n$ accordingly with k additional entries each, and then we consider the event that the $(r+k)$ -dimensional random vector $\xi_1\vec a_1+\dots+\xi_n\vec a_n$ falls into a certain quadric $\mathcal Z$ (still described by the same polynomial P) on the codimension-k affine-linear subspace $\mathcal W\subseteq \mathbb{R}^{r+k}$ given by $\mathcal W=\{\vec y \in \mathbb{R}^{r+k} : \vec y [\{r+1,\ldots,r+k\}=\vec w\}$ (this subspace $\mathcal W\subseteq \mathbb{R}^{r+k}$ corresponds to the system of equations $M\vec\xi=\vec w$ ).

This more-or-less explains how to reduce Proposition 6.1 to the setting of Theorem 4.2. The only complication is that it may not be possible to find many disjoint bases of $\mathbb{R}^{r+k}$ among the vectors $\vec a_1,\ldots,\vec a_n\in \mathbb{R}^{r+k}$ , as is necessary to apply Theorem 4.2 (in fact, the vectors $\vec a_1,\ldots,\vec a_n\in \mathbb{R}^{r+k}$ might not span $\mathbb{R}^{r+k}$ at all). In this case, we need to ‘drop to a subspace’ that is robustly spanned by most of the $\vec a_1,\ldots,\vec a_n$ (i.e., we need to identify some large subset $I\subseteq [n]$ , such that we can find many disjoint bases of $\operatorname{span}(\vec a_i:i\in I)$ among the vectors $\vec a_i$ for $i\in I$ ). This, roughly speaking, corresponds to considering only some subset of the linear forms $g^{({1})},\ldots,g^{(r+k)}$ , such that the remaining linear forms are ‘close to’ being linear combinations of these linear forms (i.e., each of the remaining linear forms agrees with such a linear combination in its coefficients of the variables $x_i$ for $i\in I$ ). The quadratic polynomial $P(\vec{y})$ considered above then needs to be changed appropriately in order to reflect these relations between the linear forms $g^{({1})},\ldots,g^{(r+k)}$ .

The following lemma encapsulates this translation discussed above, going from the setting of Proposition 6.1 to the setting of Theorem 4.2.

Then, we can find a positive integer $\ell\le r$ , a partition $[n]=I\cup S$ with $|S|\le s$ , linear forms $g^{(1)},\ldots,g^{({\ell+k})} \in \mathbb{R}[x_1,\ldots,x_n]$ , and a quadratic polynomial $P\in \mathbb{R}[y_1,\ldots,y_{\ell+k}, (x_i)_{i\in S}]$ such that the following conditions hold.

1. (i) $Q(x_1,\ldots,x_n)=P(g^{(1)}(\vec{x}),\ldots,g^{({\ell+k})}(\vec{x}), (x_i)_{i\in S})$ .

2. (ii) For each $j=1,\ldots,k$ , the coefficient vector of $g^{({\ell+j})}$ is precisely the jth row of M.

3. (iii) Writing $M'\in \mathbb{R}^{(\ell+k)\times n}$ for the $(\ell+k)\times n$ matrix whose jth row is the coefficient vector of $g^{(j)}$ for $j=1,\ldots,\ell+k$ , we have $M'[[\ell+k]\times I]\in \mathcal{H}^{(\ell+k)\times I}(s/r)$ .

4. (iv) There exist $j,j'\in [\ell]$ such that the coefficient of $y_jy_{j'}$ in P is nonzero.

We remark that the vectors $\vec a_i$ for $i\in I$ in the informal explanation above correspond to the columns of the matrix $M'[[\ell+k]\times I]$ in Lemma 6.2. Condition (iii) ensures that we can find many disjoint bases of $\mathbb{R}^{\ell+k}$ among these vectors $\vec a_i$ for $i\in I$ (recall that this is required to apply Theorem 4.2 to these vectors). Condition (iv) ensures that the polynomial P does not vanish on our entire subspace $\mathcal{W}$ (corresponding to a system of equations of the form $M\vec{\xi}=\vec{w}$ ).

First, in order to see that this minimum number $\ell$ is well defined, let us check that $\ell=r$ satisfies the conditions. Indeed, given that $\operatorname{rank} A\le r-1$ , we can write

$$ Q(\vec x)=\lambda_{1}\cdot (g^{({1})}(\vec{x}))^2+\dots+\lambda_{r-1} \cdot (g^{({r-1})}(\vec{x}))^2+g^{(r)}(\vec x)+c, $$

for some $\lambda_1,\ldots,\lambda_{r-1},c\in \mathbb{R}$ and some linear forms $g^{({1})},\ldots,g^{({r-1})},g^{(r)} \in \mathbb{R}[x_1,\ldots,x_n]$ . That is to say, defining $g^{({r+1})},\ldots,g^{({r+k})} \in \mathbb{R}[x_1,\ldots,x_n]$ to be the linear forms whose coefficient vectors are given by the rows of M as in condition (ii), we can write

$$ Q(\vec x)=P(g^{(1)}(\vec{x}),\ldots,g^{(r+k)}(\vec{x})), $$

where $P(y_1,\ldots,y_{r+k})=\lambda_1 y_1^2+\dots+\lambda_{r-1} y_{r-1}^2+y_r+c$ (formally this is a polynomial in $r+k$ variables, despite the fact that $y_{r+1},\ldots,y_{r+k}$ do not actually appear in it). Note that (ii) holds by definition, and taking $I=[n]$ and $S=\emptyset$ , condition (i) holds as well and we have $|S|=0\le s\cdot (r-r)/r$ . Thus, $\ell=r$ has the required properties, and the minimum number $\ell$ above is well defined.

Now let $\ell$ be this minimum number, and choose the partition $[n]=I\cup S$ , the linear forms $g^{(1)},\ldots,g^{({\ell+k})} \in \mathbb{R}[x_1,\ldots,x_n]$ and the quadratic polynomial P accordingly with the properties above. Note that $|S|\le s\cdot (r-\ell)/r\le s$ , and consequently $|I|\ge n-s$ . It remains to show that $\ell\ge 1$ and that conditions (iii) and (iv) hold.

Let us first show (iv). Suppose for contradiction that in the quadratic polynomial P, the coefficient of $y_jy_{j'}$ is zero for all $j,j'\in [\ell]$ . Then the quadratic part of $Q(x_1,\ldots,x_n)=P(g^{(1)}(\vec{x}),\ldots,g^{({\ell+k})}(\vec{x}), (x_i)_{i\in S})$ can be written as a linear combination of terms of the form $g^{({j})}(\vec{x})x_h$ with $j\in \{\ell+1,\ldots,\ell+k\}$ and $h\in [n]$ , and terms of the form $x_hx_{i}$ with $h\in [n]$ and $i\in S$ . By Equation (3.1), this leads to a representation of the symmetric matrix A as a linear combination of matrices of the form $\vec{v}_j\vec{e}_h^{\,{\intercal}}$ , $\vec{e}_h\vec{v}_j^{\,{\intercal}}$ , $\vec{e}_i\vec{e}_{h}^{\,{\intercal}}$ and $\vec{e}_h\vec{e}_{i}^{\,{\intercal}}$ for $j\in \{\ell+1,\ldots,\ell+k\}$ , $h\in [n]$ and $i\in S$ , where $\vec{e}_1,\ldots,\vec{e}_n$ are the standard basis vectors in $\mathbb{R}^n$ , and $\vec{v}_{\ell+1},\ldots,\vec{v}_{\ell+k}\in \mathbb{R}^n$ denote the coefficient vectors of $g^{({\ell+1})},\ldots,g^{({\ell+k})}$ (i.e., the row vectors of M, by condition (ii)). Hence the matrix $A[I\times I]$ is a linear combination of matrices of the form $\vec{v}_j[I]\vec{e}_h^{\,{\intercal}}[I]$ and $\vec{e}_h[I]\vec{v}_j^{\,{\intercal}}[I]$ for $j\in \{\ell+1,\ldots,\ell+k\}$ and $h\in I$ . Recalling that $\vec{v}_{\ell+1},\ldots,\vec{v}_{\ell+k}$ are the row vectors of the matrix M, this means that $A[I\times I]$ is a $(M[[k]\times I],M[[k]\times I])$ -perturbation of the zero matrix, contradicting our assumption. So indeed, for some monomial $y_jy_{j'}$ with $j,j'\in [\ell]$ , the coefficient of $y_jy_{j'}$ in P must be nonzero. This also automatically implies that $\ell\ge 1$ .

It now remains to show (iii), i.e., to show that for the matrix $M'\in \mathbb{R}^{(\ell+k)\times n}$ whose rows are given by the coefficient vectors of $g^{(1)},\ldots,g^{({\ell+k})}$ we have $M'[[\ell+k]\times I]\in \mathcal{H}^{(\ell+k)\times I}(s/r)$ . Assume the contrary; then there exists a nontrivial linear combination of the rows of $M'[[\ell+k]\times I]$ yielding a vector with at most $s/r$ nonzero entries. By (ii), the rows of the matrix $M'[\{\ell+1,\ldots,\ell+k\}\times [n]]$ agree with the rows of the matrix M, and we therefore have $M'[\{\ell+1,\ldots,\ell+k\}\times [n]]\in \mathcal{H}^{k\times n}(s)$ (recalling that $M\in \mathcal{H}^{k\times n}(s)$ ). Using $|S|\le s\cdot (r-\ell)/r\le s\cdot (r-1)/r$ , we can conclude that $M'[\{\ell+1,\ldots,\ell+k\}\times I]\in \mathcal{H}^{k\times I}(s-|S|)\subseteq \mathcal{H}^{k\times I}(s/r)$ , so our linear combination cannot only involve rows with indices in $\{\ell+1,\ldots,\ell+k\}$ . Thus, we may assume without loss of generality that it involves the first row of $M'[[\ell+k]\times I]$ , meaning that this first row differs in at most $s/r$ entries from some linear combination of the other rows of $M'[[\ell+k]\times I]$ . In other words, we can write $g^{({1})}$ as a linear combination of $g^{(2)},\ldots,g^{({\ell+k})}$ and $x_i$ for $i\in S\cup S'$ , for some subset $S'\subseteq [n]$ of size $|S'|\le s/r$ . But this means that $Q(x_1,\ldots,x_n)=P(g^{(1)}(\vec{x}),\ldots,g^{({\ell+k})}(\vec{x}), (x_i)_{i\in S})$ can be written as $P^*(g^{(2)}(\vec{x}),\ldots,g^{({\ell+k})}(\vec{x}), (x_i)_{i\in S\cup S'})$ for some quadratic polynomial $P^*\in \mathbb{R}[y_2,\ldots,y_{\ell+k}, (x_i)_{i\in S\cup S'}]$ . Since $|S\cup S'|\le |S|+|S'|\le s\cdot (r-\ell)/r+s/r=s\cdot (r-\ell+1)/r$ , this contradicts the minimality of $\ell$ (we now have a suitable representation of Q in terms of the $(\ell-1)+k$ linear forms $g^{(2)},\ldots,g^{({\ell+k})}$ ). So we must have $M'[[\ell+k]\times I]\in \mathcal{H}^{(\ell+k)\times I}(s/r)$ , as desired.

Using Lemma 6.2 and Theorem 4.2, we now prove Proposition 6.1.

Our plan is to show that the bound in Equation (6.1) holds even if we condition on an arbitrary outcome of $\vec \xi[S]$ (leaving only the randomness in $\vec \xi[I]$ ). For any outcome of $\vec \xi[S]$ , when plugging in $x_i=\vec{\xi}[i]$ for $i\in S$ into the polynomial $Q(\vec x)$ and the linear forms $g^{(1)}(\vec x),\ldots,g^{({\ell+k})}(\vec x)$ , we obtain a polynomial $Q_{\vec \xi[S]}(\vec x[I])$ and linear functions $g^{(1)}_{*}(\vec x[I])+c^{(1)}_{\vec \xi[S]},\ldots,g^{({\ell+k})}_{*}(\vec x[I])+c^{(\ell+k)}_{\vec \xi[S]}$ (where $g^{(1)}_{*}(\vec x[I]),\ldots,g^{({\ell+k})}_{*}(\vec x[I])$ are the linear forms obtained from $g^{(1)}(\vec x),\ldots,g^{({\ell+k})}(\vec x)$ by omitting all terms with variables $x_i$ for $i\in S$ , and where $c^{(1)}_{\vec \xi[S]},\ldots,c^{({\ell+k})}_{\vec \xi[S]}\in \mathbb{R}$ are real numbers that may depend on $\vec \xi[S]$ ) and by (i) we obtain that $Q_{\vec \xi[S]}(\vec{x}[I])=P(g^{(1)}_{*}(\vec x[I])+c^{(1)}_{\vec \xi[S]},\ldots,g^{({\ell+k})}_{*}(\vec x[I])+c^{(\ell+k)}_{\vec \xi[S]}, (\xi[i])_{i\in S})$ . For any outcome of $\vec \xi[S]$ , we can furthermore write $P(y_1+c^{(1)}_{\vec \xi[S]},\ldots,y_{\ell+k}+c^{(\ell+k)}_{\vec \xi[S]},(\xi[i])_{i\in S})=P_{\vec \xi[S]}(y_1,\ldots,y_{\ell+k})$ for a polynomial $P_{\vec \xi[S]}\in \mathbb{R}[y_1,\ldots,y_{\ell+k}]$ whose coefficients may depend on $\vec \xi[S]$ . Then we always have $Q_{\vec \xi[S]}(\vec{x}[I])=P_{\vec \xi[S]}(g^{(1)}_{\vec \xi[S]}(\vec{x}[I]]),\ldots,g^{({\ell+k})}_{\vec \xi[S]}(\vec{x}[I]))$ . Recall that the coefficient vectors of the linear forms $g^{(1)}_{*}(\vec x[I]),\ldots,g^{({\ell+k})}_{*}(\vec x[I])$ are obtained by restricting the coefficient vectors of $g^{(1)}(\vec x),\ldots,g^{({\ell+k})}(\vec x)$ to the index set I. In particular, by condition (ii), for $j=1,\ldots,k$ the coefficient vector of $g^{({\ell+j})}_{*}(\vec x[I])$ is the restriction $M[[j]\times I]$ of the jth row of M to I. We then have

\begin{align*}&\mathrm{Pr}[Q(\vec \xi)=0 \text{ and } M\vec{\xi}=\vec{w}\,|\,\vec \xi[S]]\\&\qquad =\mathrm{Pr}[Q_{\vec \xi[S]}(\vec \xi[I])=0 \text{ and } M[[k]\times I]\vec{\xi}[I]=\vec{w}_{\vec \xi[S]}\,|\,\vec \xi[S]]\\&\qquad =\mathrm{Pr}[P_{\vec \xi[S]}(g^{(1)}_{*}(\vec{\xi}[I]),\ldots,g^{({\ell+k})}_{*}(\vec{\xi}[I]))=0 \text{ and } g^{({\ell+j})}_{*}(\vec{\xi}[I])=\vec{w}_{\vec \xi[S]}[j]\text{ for }j=1,\ldots,k\,|\,\vec \xi[S]],\end{align*}

for any outcome of $\vec \xi[S]$ , where $\vec{w}_{\vec \xi[S]}=\vec{w}-M[[k]\times S]\vec{\xi}[S]\in \mathbb{R}^k$ . We wish to show that for all outcomes of $\vec \xi[S]$ , the above conditional probability is bounded by the right-hand side of Equation (6.1).

Recall that $P\in \mathbb{R}[y_1,\ldots,y_{\ell+k}, (x_i)_{i\in S}]$ is a quadratic polynomial, and by condition (iv), for some $j,j'\in [\ell]$ the coefficient of $y_j y_{j'}$ in P is nonzero. This means that the coefficient of $y_jy_{j'}$ in $P_{\vec \xi[S]}(y_1,\ldots,y_{\ell+k})\in \mathbb{R}[y_1,\ldots,y_{\ell+k}]$ is still nonzero, for any outcome of $\vec \xi[S]$ .

Since the coefficient vectors of the linear forms $g^{(1)}(\vec x),\ldots,g^{({\ell+k})}(\vec x)\in \mathbb{R}[x_1,\ldots,x_n]$ are the rows of the matrix M’ in condition (iii), the coefficient vectors of $g^{(1)}_{*}(\vec x[I]),\ldots,g^{({\ell+k})}_{*}(\vec x[I])\in \mathbb{R}[x_i: i\in I]$ are the rows of the matrix $M'[[\ell+k]\times I]$ . Now, define the vectors $\vec{a}_i\in \mathbb{R}^{\ell+k}$ , for $i\in I$ , to be the columns of the matrix $M'[[\ell+k]\times I]$ . Then for any outcome of $\vec{\xi}[I]$ , the vector $(g^{(1)}_{*}(\vec{\xi}[I]),\ldots,g^{({\ell+k})}_{*}(\vec{\xi}[I]))$ agrees with $\sum_{i\in I} \vec{\xi}[i]\vec{a}_i$ .

Furthermore, as $M'[[\ell+k]\times I]\in \mathcal{H}^{(\ell+k)\times I}(s/r)$ , by Lemma 3.2(ii) the matrix $M'[[\ell+k]\times I]$ must contain $\lceil s/(r(\ell+k))\rceil\ge \lceil s/(k+r)^2\rceil$ disjoint nonsingular $(\ell+k)\times (\ell+k)$ submatrices, so among the vectors $\vec{a}_i$ for $i\in I$ we can form $\lceil s/(k+r)^2\rceil$ disjoint bases of $\mathbb{R}^{\ell+k}$ . Consider the largest integer $t\le \lceil s/(k+r)^2\rceil$ which is divisible by $2^{\ell-1}$ , and note that $t\ge \lceil s/(k+r)^2\rceil/2\ge s/(2(k+r)^2)$ , since $\lceil s/(k+r)^2\rceil\ge 2^{3r^2}\ge 2^{\ell-1}$ .

Now, for any outcome of $\vec \xi[S]$ , we obtain

\begin{align*}&\mathrm{Pr}[Q(\vec \xi)=0 \text{ and } M\vec{\xi}=\vec{w}\,|\,\vec \xi[S]]\\&\qquad=\mathrm{Pr}\bigg[P_{\vec \xi[S]}\bigg(\sum_{i\in I}\vec \xi[i]\vec{a}_i\bigg)=0 \text{ and } \sum_{i\in I}\vec \xi[i]\vec{a}_i[\ell+j]=\vec{w}_{\vec \xi[S]}[j]\text{ for }j=1,\ldots,k\,|\,\vec \xi[S]\bigg]\\&\qquad=\mathrm{Pr}\bigg[P_{\vec \xi[S]}\bigg(\sum_{i\in I}\vec \xi[i]\vec{a}_i\bigg)=0 \text{ and } \sum_{i\in I}\vec \xi[i]\vec{a}_i\in \mathcal{W}_{\vec \xi[S]}\,|\,\vec \xi[S]\bigg]=\mathrm{Pr}\bigg[\sum_{i\in I}\vec \xi[i]\vec{a}_i\in \mathcal{Z}_{\vec \xi[S]}\,|\,\vec \xi[S]\bigg],\end{align*}

where $\mathcal{W}_{\vec \xi[S]}\subseteq \mathbb{R}^{\ell+k}$ is the $\ell$ -dimensional affine-linear subspace consisting of the points $\vec{y}\in \mathbb{R}^{\ell+k}$ with $\vec{y}[\ell+j]=\vec{w}_{\vec \xi[S]}[j]$ for $j=1,\ldots,k$ , and $\mathcal{Z}_{\vec \xi[S]}\subseteq \mathcal{W}_{\vec \xi[S]}$ is the subset of $\mathcal{W}_{\vec \xi[S]}$ given by $\mathcal{Z}_{\vec \xi[S]}=\{\vec{y}\in \mathcal{W}_{\vec \xi[S]}: P_{\vec \xi[S]}(\vec{y})=0\}$ .

We claim that for any outcome of $\vec \xi[S]$ , we have $\mathcal{Z}_{\vec \xi[S]}\subsetneq \mathcal{W}_{\vec \xi[S]}$ , i.e., $\mathcal{Z}_{\vec \xi[S]}$ is a quadric on $\mathcal{W}_{\vec \xi[S]}$ . Indeed, if we had $\mathcal{Z}_{\vec \xi[S]}= \mathcal{W}_{\vec \xi[S]}$ , then the polynomial $P_{\vec \xi[S]}$ would be identically zero on the entire subspace $\mathcal{W}_{\vec \xi[S]}\subseteq \mathbb{R}^{\ell+k}$ . Note that, on the space $\mathcal{W}_{\vec \xi[S]}$ , we can identify $P_{\vec \xi[S]}$ with some quadratic polynomial in the variables $y_{1},\ldots,y_{\ell}$ (obtained by substituting $y_{\ell+j}=\vec{w}_{\vec \xi[S]}[j]$ for $j=1,\ldots,k$ ). This polynomial has a nonzero coefficient for some monomial of the form $y_jy_{j'}$ with $j,j'\in [\ell]$ , since $P_{\vec \xi[S]}$ also has a nonzero coefficient for such a monomial. Hence the polynomial $P_{\vec \xi[S]}$ cannot vanish on the entire subspace $\mathcal{W}_{\vec \xi[S]}$ , so indeed $\mathcal{Z}_{\vec \xi[S]}\subsetneq \mathcal{W}_{\vec \xi[S]}$ .

Recalling that among the vectors $\vec{a}_i$ for $i\in I$ we can form t disjoint bases of $\mathbb{R}^{\ell+k}$ , we can now apply Theorem 4.2 (after conditioning on any outcome of $\vec \xi[S]$ ) and obtain the bound

\begin{align*}\mathrm{Pr}[Q(\vec \xi)=0 \text{ and } M\vec{\xi}=\vec{w}\,|\, \vec \xi[S]]&=\mathrm{Pr}\Biggl[\sum_{i\in I}\vec \xi[i]\vec{a}_i\in \mathcal{Z}_{\vec \xi[S]}\,\Bigg|\, \vec \xi[S]\Biggr]\\&\le \frac{2^{(\ell-1)(\ell+k)+1}}{t^{((\ell+k)-(\ell-1))/2}}\le \frac{2^{(\ell-1)\ell(k+1)+1}}{t^{(k+1)/2}}\le \frac{2^{\ell^2(k+1)}}{(s/(2(k+r)^2))^{(k+1)/2}}\\&\le \bigg(\frac{2^{3\ell^2}(k+r)^2}{s}\bigg)^{(k+1)/2}\!\!= \bigg(\frac{s}{2^{3r^2}(k+r)^2}\bigg)^{-(k+1)/2}\!,\end{align*}

as desired.

7. Robust rank inheritance

In this section, we prove Lemma 5.7 (the robust rank inheritance lemma). Actually, we will deduce Lemma 5.7 from the following somewhat simpler statement with just two matrices T,A instead of three matrices T,U,A (to deduce Lemma 5.7, we will take $r=2$ in the statement below).

1. (i) For $t=1,\ldots,s$ , the submatrix $T[[k]\times I_t]$ has rank k.

2. (ii) For $t=1,\ldots,s$ , every $(T[[k]\times I_t],0)$ -perturbation of the matrix $A[J_t\times I_t]$ has rank at least r.

Then, for a sequence of independent Rademacher random variables $\vec{\xi}=(\xi_{1},\ldots,\xi_{m})\in\{-1,1\}^{m}$ , we have

(7.1) \begin{equation}\mathrm{Pr}[T\vec{\xi}=\vec{w} \, { and } \, A\vec{\xi}-\vec{v} \, { has \, at \, most }s/6 \, { nonzero \, coordinates}]\le\bigg(\frac{s}{10^{60}(k+r)^{20}}\bigg)^{-(k+r)/2}.\end{equation}

Note that, if $k=0$ , then condition (i) is vacuous, and in condition (ii), there are no nontrivial $(T[[k]\times I_t],0)$ -perturbations. This case of Lemma 7.1 implies Lemma 2.1 (the ‘Hamming norm’ anticoncentration inequality mentioned in § 2); the formal deduction can be found at the end of this section.

Also, note that assumptions (i) and (ii) in particular imply that for all $t=1,\ldots,s$ we have

(7.2) \begin{equation}\operatorname{rank}\begin{pmatrix}A[J_t\times I_t]\\ T[[k]\times I_t]\end{pmatrix}\ge k+r.\end{equation}

Indeed, if this rank were at most $k+r-1$ , then one would be able to form a basis of the row span of the above matrix using the k rows of $T[[k]\times I_t]$ and up to $r-1$ rows of $A[J_t\times I_t]$ (recall that by assumption (i) the rows of the matrix $T[[k]\times I_t]$ are linearly independent). But then it would be possible to add linear combinations of the rows of $T[[k]\times I_t]$ to the rows of $A[J_t\times I_t]$ to obtain a matrix of rank at most $r-1$ , contradicting assumption (ii).

In order to prove Lemma 7.1 we will employ a witness-counting approach (as outlined in § 2.4), reminiscent of certain arguments of Ferber, Jain, Luh and Samotij [Reference Ferber, Jain, Luh and SamotijFJLS21]. We need to bound the probability that $T\vec{\xi}=\vec{w}$ and $A\vec{\xi}-\vec{v}$ has more than $n-s/6$ zero coordinates. Note that we can use Halász’ inequality (Theorem 3.3) to upper bound the probability that $T\vec{\xi}=\vec{w}$ and a particular set of coordinates of $A\vec{\xi}-\vec{v}$ are zero. However, it is far too wasteful to simply take a union bound over all subsets of $n-s/6$ coordinates. Instead, we consider certain types of ‘witness’ sequences $(h_1,\ldots,h_z)\in [n]^z$ of indices where the coordinates of $A\vec{\xi}-\vec{v}$ are zero. If $A\vec{\xi}-\vec{v}$ has many zero coordinates, there must be many such ‘witness’ sequences, but on the other hand we can bound the expected number of such ‘witness’ sequences by considering the probability that $(A\vec{\xi}-\vec{v})[\{h_1,\ldots,h_z\}]=\vec 0$ (and $T\vec{\xi}=\vec{w}$ ) for a given such sequence $(h_1,\ldots,h_z)$ .

The following lemma states that under the assumptions of Lemma 7.1, one can find a large submatrix of A which ‘has its rank robustly’ (the assumptions of the assumptions of Lemma 7.1 guarantee that A robustly has rank at least r, but the rank of A may actually be much larger than r, and that larger rank may be ‘fragile’). More precisely, we can find such a submatrix even within any large specified set H of rows.

Lemma 7.2. Let $r,k,s,m,n\in \mathbb{Z}$ , the matrices $T\in \mathbb{R}^{k\times m}$ and $A\in \mathbb{R}^{n\times m}$ and the sets $J_1,\ldots,J_s\subseteq [n]$ be as in Lemma 7.1. Then, for any subset $H\subseteq J_1\cup\dots\cup J_s$ of size $|H|\ge (k+r)s-(2/3)s$ , there are subsets $J\subseteq H$ and $I\subseteq [m]$ with $|J|\ge |H|-s/6$ and $|I|\ge m-s/6$ and an integer $z\ge r$ such that

(7.3) \begin{equation}\operatorname{rank}\begin{pmatrix}A[J\times I]\\ T[[k]\times I]\end{pmatrix}=\operatorname{rank}\begin{pmatrix}A[J'\times I']\\ T[[k]\times I']\end{pmatrix}=k+z,\end{equation}

for all subsets $J'\subseteq J$ and $I'\subseteq I$ of sizes $|J'|\ge |J|-s/(12z^2)$ and $|I'|\ge |I|-s/(12z^2)$ .

Proof. For every integer $z\ge 0$ , define $f(z)=\sum_{y=z+1}^{\infty} s/(12y^2)$ and note that $f(z)\le \sum_{y=1}^{\infty} s/(12y^2)\lt s/6$ for all $z\ge 0$ . Now consider the minimum integer $z\ge 0$ such that there exist subsets $J\subseteq H$ and $I\subseteq[m]$ with $|J|\ge |H|-f(z)$ and $|I|\ge m-f(z)$ and

$$ \operatorname{rank}\begin{pmatrix}A[J\times I]\\ T[[k]\times I]\end{pmatrix}\le k+z; $$

(such an integer z exists, for example $z=m$ satisfies the condition when taking $J= H$ and $I=[m]$ ). We claim that $z\ge r$ . Indeed, note that $|J|\ge |H|-f(z)\gt (k+r)s-(2/3)s-s/6=|J_1\cup\dots\cup J_s|-(5s/6)$ and $|I|\ge m-f(z)\gt m-s/6$ , so there are strictly fewer than s indices $t\in \{1,\ldots,s\}$ such that $I_t\setminus I\ne \emptyset$ or $J_t\setminus J\ne \emptyset$ . Hence, there must be some index $t\in \{1,\ldots,s\}$ with $I_t\subseteq I$ and $J_t\subseteq J$ , and we have

$$ k+z\ge \operatorname{rank}\begin{pmatrix}A[J\times I]\\ T[[k]\times I]\end{pmatrix}\ge \operatorname{rank}\begin{pmatrix}A[J_t\times I_t]\\ T[[k]\times I_t]\end{pmatrix}\ge k+r, $$

where the last step follows from the assumptions in Lemma 7.1, see Equation (7.2). So we indeed have $z\ge r\ge 1$ .

Finally, for any subsets $I'\subseteq I$ and $J'\subseteq J$ of sizes $|I'|\ge |I|-s/(12z^2)$ and $|J'|\ge |J|-s/(12z^2)$ , we have $|I'|\ge m-f(z)-s/(12z^2)=m-f(z-1)$ and $|J'|\ge |H|-f(z)-s/(12z^2)=|H|-f(z-1)$ , so by the choice of z we must have

$$ k+z\ge \operatorname{rank}\begin{pmatrix}A[J\times I]\\ T[[k]\times I]\end{pmatrix}\ge \operatorname{rank}\begin{pmatrix}A[J'\times I']\\ T[[k]\times I']\end{pmatrix}\ge k+(z-1)+1=k+z, $$

implying Equation (7.3).

We can use the robust rank submatrix guaranteed by Lemma 7.2 to find many sequences $(h_1,\ldots,h_z)$ to be used as ‘witnesses’, as follows.

Corollary 7.3. Let $r,k,s,m,n\in \mathbb{Z}$ , the matrices $T\in \mathbb{R}^{k\times m}$ and $A\in \mathbb{R}^{n\times m}$ and the sets $J_1,\ldots,J_s\subseteq [n]$ be as in Lemma 7.1. Then for any subset $H\subseteq J_1\cup\dots\cup J_s$ of size $|H|\ge (k+r)s-(2/3)s$ , there is some integer $z\ge r$ such that there are at least $s^z/(12z^2)^z$ sequences $(h_1,\ldots,h_z)\in H^z$ with

(7.4) \begin{equation}\begin{pmatrix}A[\{h_{1},\ldots,h_{z}\}\times[m]]\\T\end{pmatrix}\in \mathcal H^{(k+z)\times m}(s/(12z^2)).\end{equation}

Proof. Let $J\subseteq H$ and $I\subseteq [m]$ and $z\ge r$ be as in Lemma 7.2. We first claim that there are at least $s^z/(12z^2)^z$ sequences $(h_1,\ldots,h_z)\in J^z\subseteq H^z$ such that

(7.5) \begin{equation} \operatorname{row-span}\begin{pmatrix}A[\{h_{1},\ldots,h_{z}\}\times I]\\ T[[k]\times I] \end{pmatrix} =\operatorname{row-span}\begin{pmatrix}A[J\times I]\\ T[[k]\times I] \end{pmatrix}. \end{equation}

Indeed, the matrix on the right-hand side has rank $k+z$ by Equation (7.3), and the rows of $T[[k]\times I]$ are linearly independent by assumption (i) in Lemma 7.1 (since $|I|\ge m-s/6$ , there needs to be at least one index $t\in \{1,\ldots,s\}$ with $I_t\subseteq I$ , meaning that $\operatorname{rank} T[[k]\times I]=T[[k]\times I_t]=k$ ). So we can form a basis of the row span of the matrix on the right-hand side by taking the rows of $T[[k]\times I]$ and adding, one by one, z different rows of $A[J\times I]$ . By Equation (7.3), at every step we have at least $s/(12z^2)$ choices for a new row to add to our basis (indeed, the index set $J'\subseteq J$ of all rows in the span of the already selected basis elements must have size $|J'|<|J|-s/(12z^2)$ , since this span has dimension less than $k+z$ and so otherwise we would have a contradiction to Equation (7.3) with $I'=I$ ). Thus, there are indeed at least $s^z/(12z^2)^z$ choices for the sequence $(h_1,\ldots,h_z)\in J^z$ of indices of the rows of $A[J\times I]$ selected in this process. For each such sequence Equation (7.5) holds.

It remains to show that every sequence $(h_1,\ldots,h_z)\in J^z$ satisfying Equation (7.5) also satisfies Equation (7.4). So, consider any $(h_1,\ldots,h_z)\in J^z$ satisfying Equation (7.5). To show Equation (7.4), it suffices to show that

(7.6) \begin{equation} \begin{pmatrix}A[\{h_{1},\ldots,h_{z}\}\times I]\\ T[[k]\times I] \end{pmatrix}\in \mathcal H^{(k+z)\times I}(s/(12z^2)). \end{equation}

Suppose the contrary; then there exists a subset $I'\subseteq I$ of size $|I'|\ge |I|-s/(12z^2)$ such that

$$ \operatorname{rank} \begin{pmatrix}A[\{h_{1},\ldots,h_{z}\}\times I']\\ T[[k]\times I'] \end{pmatrix}<k+z. $$

But now by Equation (7.5) we have

$$ \operatorname{row-span}\begin{pmatrix}A[J\times I']\\ T[[k]\times I'] \end{pmatrix} =\operatorname{row-span}\begin{pmatrix}A[\{h_{1},\ldots,h_{z}\}\times I']\\ T[[k]\times I'] \end{pmatrix} , $$

and hence

$$ \operatorname{rank} \begin{pmatrix}A[J\times I']\\ T[[k]\times I'] \end{pmatrix} =\operatorname{rank} \begin{pmatrix}A[\{h_{1},\ldots,h_{z}\}\times I']\\ T[[k]\times I'] \end{pmatrix}<k+z, $$

contradicting Equation (7.3). So we indeed have Equation (7.6), as desired.

Now we prove Lemma 7.1. We need two slightly different implementations of our witness-counting strategy described above, with different conditions for the ‘witness’ sequences $(h_1,\ldots,h_z)$ we are counting. In the case where A ‘robustly has very high rank’, we can get away with rather crude arguments using Odlyzko’s lemma. In the complementary case, we need to use Corollary 7.3 and Halász’ inequality (we cannot simply use Corollary 7.3 and Halász’ inequality in both cases, because the conclusions of Lemma 7.2 and Corollary 7.3 are very weak when z is very large).

Proof of Lemma 7.1. LetFootnote ¹¹ $L=\lceil 2(k+r)\log s\rceil$ , let $J_*=J_1\cup\dots\cup J_s\subseteq [n]$ and note that $|J_*|=(k+r)s$ . Furthermore note that (7.1) trivially holds if $s\le 10^{60}(k+r)^{20}$ , so we may assume that $s \ge 10^{60}(k+r)^{20}\ge 10^{60}$ and hence $\log s\le s^{1/20}$ . Then we in particular have

(7.7) \begin{equation}L=\lceil 2(k+r)\log s\rceil\le 4(k+r)\cdot \log s\le \frac{s^{1/20}}{250}\cdot s^{1/20}=\frac{s^{1/10}}{250}.\end{equation}

Step 1: the robust high-rank case. First, suppose that for every subset $J'\subseteq J_*$ with $|J'|\ge |J_*|-s/2$ , we have $\operatorname{rank} A[J'\times [m]]\ge L$ .

Then, for every subset $S\subseteq J_*$ of size $|S|<L$ , the row span of $A[S\times [m]]$ has dimension less than L and so it can contain at most $|J_*|-s/2$ rows of the matrix $A[J_*\times [m]]$ . This means that there are at least $s/2$ indices $h\in J_*$ such that $\operatorname{rank} A[(S\cup \{h\})\times [m]]=\operatorname{rank} A[S\times [m]]+1$ . Let $H_S\subseteq J_*$ denote the set of the $\lceil s/2\rceil$ smallest such indices $h\in J_*$ .

For our double-counting argument, we will consider sequences $(h_{1},\ldots,h_{L})\in J_*^L$ with $h_{t}\in H_{\{h_1,\ldots,h_{t-1}\}}$ for $t=1,\ldots,L$ . For every such sequence, we have $\operatorname{rank} A[\{h_{1},\ldots,h_{t}\}\times [m]]=\operatorname{rank} A[\{h_{1},\ldots,h_{t-1}\}\times [m]]+1$ for $t=1,\ldots,L$ and hence $\operatorname{rank} A[\{h_{1},\ldots,h_{L}\}\times [m]]=L$ . Furthermore, note that the total number of such sequences is exactly $\lceil s/2\rceil^L$ (since after choosing $h_1,\ldots,h_{t-1}$ we have exactly $|H_{\{h_1,\ldots,h_{t-1}\}}|=\lceil s/2\rceil$ choices for $h_t$ ).

We claim that for every outcome of $\vec{\xi}\in \{-1,1\}^m$ such that $A\vec{\xi}-\vec{v}$ has at most $s/6$ nonzero coordinates, there are at least $(s/3)^L$ sequences $(h_{1},\ldots,h_{L})\in J_*^L$ with $h_{t}\in H_{\{h_1,\ldots,h_{t-1}\}}$ for $t=1,\ldots,L$ , such that $(A\vec{\xi}-\vec{v})[\{h_1,\ldots,h_z\}]=\vec 0$ . Indeed, choosing $h_{1},\ldots,h_{L}$ one at a time, at every step we need to choose $h_{t}\in H_{\{h_1,\ldots,h_{t-1}\}}$ with $(A\vec{\xi}-\vec{v})[h_{t}]=0$ . At most $s/6$ of the $\lceil s/2\rceil$ elements of $H_{\{h_1,\ldots,h_{t-1}\}}$ fail this condition, so we indeed have at least $\lceil s/2\rceil-s/6\ge s/3$ choices at every step and hence there are indeed at least $(s/3)^L$ such sequences $(h_{1},\ldots,h_{L})\in J_*^L$ .

Thus, the expected number of sequences $(h_{1},\ldots,h_{L})\in J_*^L$ with $h_{t}\in H_{\{h_1,\ldots,h_{t-1}\}}$ for $t=1,\ldots,L$ such that $(A\vec{\xi}-\vec{v})[\{h_1,\ldots,h_z\}]=\vec 0$ is at least $\mathrm{Pr}[A\vec{\xi}-\vec{v}\text{ has at most }s/6\text{ nonzero coordinates}]\cdot (s/3)^L$ . On the other hand, for every given sequence $(h_{1},\ldots,h_{L})\in J_*^L$ with $h_{t}\in H_{\{h_1,\ldots,h_{t-1}\}}$ for $t=1,\ldots,L$ , having $(A\vec{\xi}-\vec{v})[\{h_1,\ldots,h_z\}]=\vec 0$ is equivalent to $A[\{h_{1},\ldots,h_{L}\}\times [m]]\vec{\xi}=\vec{v}[\{h_{1},\ldots,h_{L}\}]$ and by Odlyzko’s lemma (Lemma 3.7), this happens with probability at most $2^{-L}$ (recalling that $\operatorname{rank} A[\{h_{1},\ldots,h_{L}\}\times [m]]=L$ ). Hence the expected number of such sequences $(h_{1},\ldots,h_{L})$ is at most $\lceil s/2\rceil^L\cdot 2^{-L}$ . Thus, we obtain

\begin{align*} \mathrm{Pr}[A\vec{\xi}-\vec{v}\text{ has at most }s/6\text{ nonzero coordinates}]\le \frac{\lceil s/2\rceil^L\cdot 2^{-L}}{(s/3)^L} & \le \frac{(5s/9)^L\cdot 2^{-L}}{(s/3)^L}=(5/6)^L \\& \le s^{-(k+r)/2},\end{align*}

recalling that $L=\lceil 2(k+r)\log s\rceil$ and using that $(5/6)^4<1/2$ . This in particular implies Equation (7.1).