We show that on a $\sigma $-finite measure-preserving system $X = (X,\nu , T)$, the non-conventional ergodic averages

$$ \begin{align*} \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x) \end{align*} $$

$f \in L^{p_1}(X)$

$g \in L^{p_2}(X)$

$1/p_1 + 1/p_2 \leq 1$

$2$

$\Lambda $

$1$

$\mu $

Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form

$$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$

where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$, $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$, we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$.

We study optimal dimensionless inequalities

\begin{equation*} \|f\|_{\textrm{U}^k} \leqslant \|f\|_{\ell^{p_{k,n}}} \end{equation*}

that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates

\begin{equation*} \|\mathbb{1}_A\|_{\textrm{U}^k}^{2^k}\leqslant |A|^{t_{k,n}} \end{equation*}

that hold for all subsets A of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the article:

• • an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao,

• • two-sided asymptotic estimates for $t_{k,n}$ as $n\to\infty$ for a fixed $k\geqslant2$, which generalize a theorem by Shao, and

• • a precise asymptotic formula for $t_{k,n}$ as $k\to\infty$ for a fixed $n\geqslant2$.

Let $K = \mathbf {R}$ or $\mathbf {C}$. An n-element subset A of K is a $B_h$-set if every element of K has at most one representation as the sum of h not necessarily distinct elements of A. Associated with the $B_h$-set $A = \{a_1,\ldots , a_n\}$ are the $B_h$-vectors $\mathbf {a} = (a_1,\ldots , a_n)$ in $K^n$. This article proves that “almost all” n-element subsets of K are $B_h$-sets in the sense that the set of all $B_h$-vectors is a dense open subset of $K^n$.

Ruzsa asked whether there exist Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and 4-term arithmetic progression (4-AP) density at most $\alpha^C$, for arbitrarily large C. Gowers constructed Fourier uniform sets with density $\alpha$ and 4-AP density at most $\alpha^{4+c}$ for some small constant $c \gt 0$. We show that an affirmative answer to Ruzsa’s question would follow from the existence of an $N^{o(1)}$-colouring of [N] without symmetrically coloured 4-APs. For a broad and natural class of constructions of Fourier-uniform subsets of $\mathbb Z/N\mathbb Z$, we show that Ruzsa’s question is equivalent to our arithmetic Ramsey question.

We prove analogous results for all even-length APs. For each odd $k\geq 5$, we show that there exist $U^{k-2}$-uniform subsets of $\mathbb Z/N\mathbb Z$ with density $\alpha$ and k-AP density at most $\alpha^{c_k \log(1/\alpha)}$. We also prove generalisations to arbitrary one-dimensional patterns.

We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(\!-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. We also prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some absolute constant $c \gt 0$. This improves on the previously best known upper bound $O(n^2(\log\log n)^{-c})$, coming from Shkredov’s work on the corners theorem.

Green and Tao’s arithmetic regularity lemma and counting lemma together apply to systems of linear forms which satisfy a particular algebraic criterion known as the ‘flag condition’. We give an arithmetic regularity lemma and counting lemma which apply to all systems of linear forms.

In this article, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine equations. Among many new results, we highlight our main contributions as follows. First, we show that if a nontrivial shift of a multiplicative subgroup G contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, the largest size of a set such that each pairwise product of its elements is n less than a kth power, refining the recent result of Dixit, Kim, and Murty. One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest. We also make significant progress toward a conjecture of Sárközy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes p, the set $\{x^2-1: x \in {\mathbb F}_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in ${\mathbb F}_p$ non-trivially.

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs (i.e., $x,y\in {\mathbb N}$ such that $x^2\pm y^2=z^2$ for some $z\in {\mathbb N}$). We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group G from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$, for some infinite sequence $(b_n)_{n\in \mathbb {N}}$ and some $t\in G$. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13].

We establish the restricted sumset analog of the celebrated conjecture of Sárközy on additive decompositions of the set of nonzero squares over a finite field. More precisely, we show that if $q>13$ is an odd prime power, then the set of nonzero squares in $\mathbb {F}_q$ cannot be written as a restricted sumset $A \hat {+} A$, extending a result of Shkredov. More generally, we study restricted sumsets in multiplicative subgroups over finite fields as well as restricted sumsets in perfect powers (over integers) motivated by a question of Erdős and Moser. We also prove an analog of van Lint–MacWilliams’ conjecture for restricted sumsets, which appears to be the first analogue of Erdős--Ko–Rado theorem in a family of Cayley sum graphs.

An extension of Szemerédi’s theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and Tcaciuc. Via a novel version of Furstenberg’s correspondence principle, which should be of independent interest, we show that our Szemerédi theorems can be deduced from a general transverse multiple recurrence theorem, which we establish using a recent work of Austin [Non-conventional ergodic averages for several commuting actions of an amenable group. J. Anal. Math. 130 (2016), 243–274].

Let p be a prime, $q=p^n$, and $D \subset \mathbb {F}_q^*$. A celebrated result of McConnel states that if D is a proper subgroup of $\mathbb {F}_q^*$, and $f:\mathbb {F}_q \to \mathbb {F}_q$ is a function such that $(f(x)-f(y))/(x-y) \in D$ whenever $x \neq y$, then $f(x)$ necessarily has the form $ax^{p^j}+b$. In this notes, we give a sufficient condition on D to obtain the same conclusion on f. In particular, we show that McConnel’s theorem extends if D has small doubling.

In this paper, we study ergodic $\mathbb {Z}^r$-actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions, there are always directions which expand significantly a given measurable set with positive measure. Among other things, we use this result to prove that the set of volumes of all r-simplices with vertices in a set with positive upper density must contain an infinite arithmetic progression, thus showing a discrete density analogue of a classical result by Graham.

Let $P_1, \ldots , P_m \in \mathbb {K}[\mathrm {y}]$ be polynomials with distinct degrees, no constant terms and coefficients in a general local field $\mathbb {K}$. We give a quantitative count of the number of polynomial progressions $x, x+P_1(y), \ldots , x + P_m(y)$ lying in a set $S\subseteq \mathbb {K}$ of positive density. The proof relies on a general $L^{\infty }$ inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex and p-adic analysis.

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

For $E \subset \mathbb {N}$, a subset $R \subset \mathbb {N}$ is E-intersective if for every $A \subset E$ having positive relative density, $R \cap (A - A) \neq \varnothing $. We say that R is chromatically E-intersective if for every finite partition $E=\bigcup _{i=1}^k E_i$, there exists i such that $R\cap (E_i-E_i)\neq \varnothing $. When $E=\mathbb {N}$, we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when $E = \mathbb {P}$, the set of primes, or other sparse subsets of $\mathbb {N}$. Among other things, we prove the following: (1) the set of shifted Chen primes $\mathbb {P}_{\mathrm {Chen}} + 1$ is both intersective and $\mathbb {P}$-intersective; (2) there exists an intersective set that is not $\mathbb {P}$-intersective; (3) every $\mathbb {P}$-intersective set is intersective; (4) there exists a chromatically $\mathbb {P}$-intersective set which is not intersective (and therefore not $\mathbb {P}$-intersective).

In this paper, we prove that the bound

\begin{equation*}\max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}}\end{equation*}

$A \subset \mathbb R$

\begin{equation*}\max \{ |16A|, |A^{(16)}| \} \gg |A|^{\frac{3}{2} + c},\end{equation*}

$c\gt 0$

$\mathbb R$

$3/2$

\begin{equation*}|AA| \leq K|A| \implies \,\forall d \in \mathbb R \setminus \{0 \}, \,\, |\{(a,b) \in A \times A : a-b=d \}| \ll K^C |A|^{\frac{2}{3}-c^{\prime}},\end{equation*}

$c,C \gt 0$

We introduce a new concept of rank – relative rank associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case, we obtain a new proof of the results in [21]). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is of high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry, and algebra. For example, we prove that any collection of polynomials $\mathcal P=(P_i)_{i=1}^c$ of degrees $\le d$ in a polynomial ring over an algebraically closed field of characteristic $>d$ is contained in an ideal $\mathcal I({\mathcal Q})$, generated by a collection ${\mathcal Q}$ of polynomials of degrees $\le d$ which form a regular sequence, and ${\mathcal Q}$ is of size $\le A c^{A}$, where $A=A(d)$ is independent of the number of variables.

We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory and the geometry of numbers. We give applications of these results to new bounds on correlations between Salié sums and to a new equidistribution estimate for the set of modular roots of primes.