2. Proof of the abstract result

Take a non-empty open $W\subset Y$ , which, since Y is locally compact, we can assume to be relatively compact. We will inductively construct a nested sequence of open sets $U_\ell \subseteq U_0 {\, \stackrel{\mathrm{def}}{=}\, } {W}$ , an increasing sequence $T_\ell \to \infty$ , and a sequence of indices $i_\ell$ so that for all $\ell \in \mathbb{N}$ :

1. (i) $\varnothing\neq \overline{ U_{\ell}}\subseteq U_{\ell-1}$ ;

2. (ii) $ L_{i_\ell} \cap U_\ell \ne \varnothing$ and ${U_\ell \cap {R_\ell} =\varnothing}$ ;

3. (iii) for all $1\leqslant k \leqslant \ell$ there exists $s = s(k,\ell)\in \mathcal{I}$ such that $d_{k,s(k,\ell)}\equiv 0$ on $\varphi_{k}(L_{i_\ell})$ and $h_{k,s(k,\ell)} \leqslant T_\ell$ ; and

4. (iv) for all $1 \leqslant k \leqslant \ell-1$ and all $x \in U_{\ell},$ $d_{k,s(k,\ell-1)}\big(\varphi_k(x)\big) < f_k(T_{\ell})$ , where $s(k,\ell-1)$ is as in (iii) (for $\ell-1$ ).

Step 1: Sufficiency. Let us first check that this is sufficient for part (a) of the theorem. First observe that (i) and the relative compactness of W imply that $ \bigcap\limits_{\ell \in \mathbb{N}} \overline{ U_\ell} $ is non-empty. We claim that

$$x \in \bigcap\limits_{\ell \in \mathbb{N}} \overline{ U_\ell} \implies x \forall\,k\in\mathbb{N}, \quad \text{$\varphi_k(x)$ is $(\mathcal{X}_k,f_k)$-{uniform}. } $$

Indeed, for any k take $T\geqslant T_{k}$ , and let $\ell$ be such that $T_\ell\leqslant T < T_{\ell+1}$ ; then clearly $\ell \geqslant k$ . Take $s = s(k,\ell)$ . Then $h_{k,s} \leqslant T_\ell\leqslant T $ by (iii), and by (iv) and since $f_k$ is non-increasing and $x \in U_{\ell+1}$ , we have that

$$ {d_{k,s}\big(\varphi_k(x)\big) < f_k({T_{\ell+1}})\leqslant f_k(T).}$$

Therefore $\varphi_k(x)$ is $(\mathcal{X}_k,f_k)$ -uniform. Also from (ii) it follows that $x\notin \bigcup_{\ell} R_\ell$ . Thus x belongs to the set (1.11), which implies its density and finishes the proof of (a).

Step 2: Base case of induction. By the total density of $\mathcal{L}$ relative to $\mathcal{R}$ there exists $i_1 $ so that

$$L_{i_1} \cap {W} \ne \varnothing \quad \text{and} \quad L_{i_1}\not\subset R_1.$$

Since $\mathcal{L}$ is aligned with $\mathcal{X}_1$ via $\varphi_1$ , there exists $s = s(1,1)$ such that $\varphi_1(L_{i_1}) $ is contained in ${d_{1,s}^{-1}(\{0\})}$ . Choose $T_1$ so that $T_1 > %H_1(A){h_{1,s}}$ , and let $z \in L_{i_1} \cap {W}$ . Choose a neighborhood $\hat{U}_1$ of z so that $\overline{\hat{U}_1} \subset{W}$ . Since $\mathcal{L}$ respects $R_1$ and the latter is closed, there is an open set $U_1 \subset\hat{U}_1$ such that $L_{i_1} \cap U_1 \neq \varnothing$ and $U_1 \cap{R_1} = \varnothing.$ This choice ensures that (ii) holds; (iii) holds by the choice of s and $T_1$ , and (i) holds since $U_1 \subset \hat{U}_1$ and $U_0 ={W}$ . Item (iv) is vacuous for $\ell=1$ . This completes the base case.

Step 3: Inductive step. Assume that we have $U_\ell, T_\ell, i_\ell$ satisfying the inductive hypotheses. In view of (1.8) and (ii), there exists $i_{\ell+1}$ so that

\[L_{i_{\ell+1}}\cap L_{i_\ell} \cap U_\ell \ne\varnothing \quad\text{and}\quad L_{i_{\ell+1}} \not\subset R_{\ell+1}.\]

Since for any $k\in\mathbb{N}$ the collection $\mathcal{L}$ is aligned with $\mathcal{X}_k$ via $\varphi_k$ , for each $k = 1, \ldots, \ell+1$ there is $s(k,\ell+1)\in \mathcal{I}$ such that $\varphi_k(L_{i_{\ell+1}}) \subset {d_{k,s(k,\ell+1)}^{-1}(\{0\})}$ . Let

$$T_{\ell+1} > \max (T_\ell \cup \{h_{k,s(k,\ell+1)} : k=1, \ldots, \ell+1\} ),$$

and let $z \in L_{i_{\ell+1}} \cap L_{i_\ell} \cap U_\ell $ . Since for $k = 1, \ldots, \ell$ the functions ${d_{k,s(k,\ell)}} \circ \varphi_k$ are continuous and vanish at $z \in L_{i_\ell}$ , there is a neighborhood $\hat{U}_{\ell+1}$ of z such that $\overline{\hat{U}_{\ell+1} }\subset U_\ell$ and

$${d_{k,s(k,\ell)}}\big(\varphi_k(y)\big) < f_k(T_{\ell+1})\quad \text{for all }y \in\hat{U}_{\ell+1}.$$

Again, since $R_{\ell+1}$ is closed and is respected by $\mathcal{L}$ , it follows that there is an open set $U_{\ell+1} \subset\hat{U}_{\ell+1}$ such that

$$L_{i_{\ell+1}} \cap U_{\ell+1} \neq \varnothing\quad\text{and}\quad U_{\ell+1} \cap R_{\ell+1} = \varnothing.$$

This choice, and the inductive hypothesis, ensure that (ii) holds for $\ell+1$ ; (iii) holds for $\ell+1$ by the choice of $s(k,\ell+1)$ and $T_{\ell+1}$ , and (i) and (iv) hold for $\ell+1$ since $U_{\ell+1} \subset \hat{U}_{\ell+1}$ . This finishes the proof of (a).

Step 4: Part (b). The second part of Theorem 1.5 follows easily from part (a). Indeed, arguing by contradiction, assume that the set (1.11) consists of countably many points $\mathcal{R}_0{\, \stackrel{\mathrm{def}}{=}\, } \{y_j: j\in\mathbb{N}\}$ . Then one can replace $\mathcal{R}$ with $\mathcal{R}\cup \mathcal{R}_0$ which will still satisfy (1.8) and be respected by $\mathcal{L}$ (here we use the fact that every $L\in\mathcal{L}$ is either connected or has no isolated points). Thus the set (1.11) with this new choice of $\mathcal{R}$ will be empty, contradicting Theorem 1.5(a).

3. Simultaneous approximation

In this section we consider the standard Diophantine system $\mathcal{X}_{m,n}$ as in (1.6). We are going to apply Theorem 1.5 with $\varphi_k = \operatorname{Id}$ for each k and with the collection $\mathcal{L}$ parametrized by the direct product of $\mathbb{Z}^m$ and $\mathbb{Z}^n\smallsetminus\{0\}$ . Namely we will consider the collection

(3.1) \begin{equation}{\begin{aligned} \mathcal{L} {\, \stackrel{\mathrm{def}}{=}\, } \{L_{\textbf{p},\textbf{q}} : \textbf{q} \in\mathbb{Z}^n\smallsetminus\{0\}, \ \textbf{p}\in\mathbb{Z}^m\} \quad\text{where }L_{\textbf{p},\textbf{q}} {\, \stackrel{\mathrm{def}}{=}\, } \{A\in M_{m,n}(\mathbb{R}): A\textbf{q} = \textbf{p}&\}.\end{aligned}}\end{equation}

Let us prove the following proposition.

Proposition 3.1. Let $m,n\in\mathbb{N}$ with $n > 1$ , and let Y be a non-empty open subset of $M_{m,n}(\mathbb{R})$ . Then the collection

(3.2) \begin{equation}{\{L_{\textbf{p},\textbf{q}} \cap Y: \textbf{q} \in\mathbb{Z}^n\smallsetminus\{0\}, \textbf{p}\in\mathbb{Z}^m\}}\end{equation}

is totally dense.

Proof. Clearly $M_{m\times n}(\mathbb{Q})$ is dense in $M_{m,n}(\mathbb{R})$ , and, furthermore, $L_{\textbf{p},\textbf{q}}\cap M_{m\times n}(\mathbb{Q})$ is dense in $L_{\textbf{p},\textbf{q}}$ for any $\textbf{q} \in\mathbb{Z}^n\smallsetminus\{0\}$ and $\textbf{p}\in\mathbb{Z}^m$ . Moreover, any $B\in M_{m\times n}(\mathbb{Q})$ lies in an $L_{\textbf{p},\textbf{q}} $ for some $\textbf{q} \in\mathbb{Z}^n\smallsetminus\{0\}$ and $\textbf{p}\in\mathbb{Z}^m$ . This implies (1.7). Now fix $(\textbf{p}_0,\textbf{q}_0) \in\mathbb{Z}^m\times(\mathbb{Z}^n\smallsetminus\{0\})$ such that $L_{\textbf{p}_0,\textbf{q}_0}\cap Y\ne\varnothing$ , and choose an open $W \subset {Y}$ which intersects $L_{\textbf{p}_0,\textbf{q}_0}$ non-trivially. Then we pick an arbitrary

$$B\in M_{m\times n}(\mathbb{Q})\cap L_{\textbf{p}_0,\textbf{q}_0}\cap W,$$

we let $E_B$ be the union of all the subspaces $L_{\textbf{p},\textbf{q}}$ containing B (this clearly includes $L_{\textbf{p}_0,\textbf{q}_0}$ ), and we write

$$ \begin{aligned} E_B &=\, \bigcup_{(\textbf{p},\textbf{q}) \in\mathbb{Z}^m\times(\mathbb{Z}^n\smallsetminus\{0\}): B\textbf{q} = \textbf{p}}\{A\in M_{m,n}(\mathbb{R}): A\textbf{q} = \textbf{p}\}\\ &=\, \bigcup_{\textbf{q} \in\mathbb{Z}^n\smallsetminus\{0\}, \, B\textbf{q}\in\mathbb{Z}^m}\{A\in M_{m,n}(\mathbb{R}): A\textbf{q} = B\textbf{q}\} \\ &= B + \bigcup_{\textbf{q} \in\mathbb{Z}^n\smallsetminus\{0\}, \, B\textbf{q}\in\mathbb{Z}^m}\{C\in M_{m,n}(\mathbb{R}): C\textbf{q} = 0\}. \end{aligned} $$

Note that the set of $\textbf{q} \in\mathbb{Z}^n$ such that $B\textbf{q}\in\mathbb{Z}^m$ contains $N\mathbb{Z}^n$ for some $N\in\mathbb{N}$ . Therefore one has

$${E_B} \supset B + \{C\in M_{m\times n}(\mathbb{Q}): \operatorname{rank} C < n\}.$$

If $n>m$ it is clear that $E_B$ contains $M_{m\times n}(\mathbb{Q})$ , which readily implies the total density of the collection (3.2) in an even stronger form: namely, that the union in (1.9) is dense in $M_{m,n}(\mathbb{R})$ .

In general we see that $\overline{E_B} = B + R_{<n}$ , where

$$R_{<n} {\, \stackrel{\mathrm{def}}{=}\, } \{C\in M_{m,n}(\mathbb{R}): \operatorname{rank} C < n\},$$

which is a proper algebraic subvariety of $M_{m,n}(\mathbb{R})$ if $n \leqslant m$ . Now let us consider the union of the sets $E_B$ over all $B\in M_{m\times n}(\mathbb{Q})\cap L_{\textbf{p}_0,\textbf{q}_0}\cap W$ , and then take the closure, which is easily seen to have the following form:

$$\overline{\bigcup_{B\in M_{m\times n}(\mathbb{Q})\cap L_{\textbf{p}_0,\textbf{q}_0}\cap W} E_B} = \{B\in \overline{W}: B\textbf{q}_0 = \textbf{p}_0\} + R_{<n}.$$

If $n=1$ then $E_B = \{B\}$ , and the above closure coincides with $ {L_{\textbf{p}_0,\textbf{q}_0}}\cap \overline{W}$ , and hence there is no total density. Now suppose that $1 < n \leqslant m$ , and let $D\in\operatorname{GL}_n(\mathbb{R})$ be such that $\textbf{q}_0 = D\mathbf{e}_n$ . Then one can write

$$ \begin{aligned} \{B\in \overline{W}: B\textbf{q}_0 = \textbf{p}_0\} + R_{<n} &= \{B\in \overline{W}: BD\mathbf{e}_n = \textbf{p}_0\} + R_{<n} \\ &= \big(\{A\in \overline{W}{D}: A\mathbf{e}_n = \textbf{p}_0\} + R_{<n} \big)D^{-1}, \end{aligned}$$

since $R_{<n}$ is invariant under right-multiplication by invertible matrices. Thus it suffices to prove the claim under the assumption that $\textbf{q} = \mathbf{e}_n$ , that is, to show that for any non-empty open $W\subset M_{m,n}(\mathbb{R})$ and any $\textbf{p}_0\in\mathbb{Z}^m$ , the set $\{A\in \overline{W}: A\mathbf{e}_n = \textbf{p}_0\} + R_{<n} $ contains some open neighborhood V of $L_{\textbf{p}_0,\mathbf{e}_n} \cap W$ . For this it will be convenient to write the elements of $M_{m,n}(\mathbb{R})$ in a column-vector notation. Namely, we can write

$$L_{\textbf{p}_0,\mathbf{e}_n}\cap W = \big\{[\begin{matrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{n-1}& \textbf{p}_0\end{matrix}]: [\begin{matrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{n-1}\end{matrix}]\in W'\big\}$$

for some non-empty open subset W’ of $M_{m\times(n-1)}$ . Then it becomes clear that we can take

$$V {\, \stackrel{\mathrm{def}}{=}\, } \big\{[\begin{matrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{n-1}& \mathbf{b}\end{matrix}]: [\begin{matrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{n-1}\end{matrix}]\in W',\ \mathbf{b}\in\mathbb{R}^m\big\},$$

simply because any matrix $[\begin{matrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{n-1}& \mathbf{b}\end{matrix}]$ with $[\begin{matrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{n-1}\end{matrix}]\in W'$ can be written as

$$ [\begin{matrix} \mathbf{a}_1 & \cdots & \mathbf{a}_{n-1}& \textbf{p}_0\end{matrix}] + [\begin{matrix} \textbf{0} & \cdots & \textbf{0}& \mathbf{b} - \textbf{p}_0\end{matrix}]\in (L_{\textbf{p}_0,\mathbf{e}_n}\cap W) + R_{<n}.$$

This finishes the proof.

We can now use the above proposition to furnish the following proof.

More generally, one can strengthen Theorem 1.2(a) using the fact that $\mathcal{L}$ respects closed analytic submanifolds of $M_{m,n}(\mathbb{R})$ . Recall that ${\Psi}: U\to \mathbb{R}^k$ , where U is an open subset of $\mathbb{R}^d$ with $d \leqslant k$ , is called a real analytic immersion if it is injective, each of its coordinate functions ${\Psi}_i :U\to \mathbb{R} \ (i=1, \ldots, k)$ is infinitely differentiable, the Taylor series of each ${\Psi}_i$ converges in a neighborhood of every $\mathbf{x} \in U$ , and the derivative mapping $d_{\mathbf{x}}{\Psi}: \mathbb{R}^d \to \mathbb{R}^k$ has rank d. By a d-dimensional real analytic submanifold of $\mathbb{R}^k$ we mean a subset $Y \subset \mathbb{R}^k$ such that for every $\mathbf{y} \in Y$ there is a neighborhood $V\subset \mathbb{R}^k$ containing $\mathbf{y}$ , an open set $U \subset \mathbb{R}^d$ , and a real analytic immersion ${\Psi}: U \to \mathbb{R}^k$ such that ${V} \cap {Y} = {\Psi}({U}).$

The crucial property, which distinguishes real analytic submanifolds from smooth manifolds and follows easily from the definitions, is given in the following lemma.

To make the paper self-contained we provide the proof of this lemma.

Let $p \in \overline{W} \smallsetminus W$ . Since the closure is taken in L we have $p \in L$ , and since R is closed we have $p \in R$ . By the real analyticity of L and R, there are open subsets $U_1 \subset \mathbb{R}^{d_1}, \, U_2 \subset \mathbb{R}^{d_2}$ and real analytic immerions $\Psi_i: U_i \to \mathbb{R}^k$ such that $\Psi_1$ parametrizes a neighborhood of p in L, and $\Psi_2$ parametrizes a neighborhood of p in R. For $i=1,2$ , let $p_i \in U_i$ such that $\Psi_i(p_i) = p$ , and let $W_i = \Psi_i^{-1}(W)$ . Then $p_1$ belongs to the closure of $W_1$ in $U_1$ , and $W_1$ is open in $U_1$ ; thus by the uniqueness of analytic continuation (see [Reference Krantz and ParksKP92, 1.2] or [Reference Kobayashi and NomizuKN63, VI.6]), $\Psi_i$ is uniquely determined in a neighborhood of p by $\Psi_1|_{W_1}$ . In particular, L is uniquely determined in a neighborhood of p by W.

By the inverse function theorem for real analytic immersions (see [Reference Krantz and ParksKP92, 1.8]), there is a real analytic inverse $\Psi_2^{-1} $ to $\Psi_2$ . In an abuse of notation, denote by $\mathbb{R}^{d_1}$ the coordinate plane in $\mathbb{R}^{d_2}$ defined as

$$\mathbb{R}^{d_1} {\, \stackrel{\mathrm{def}}{=}\, } \{(x_1, \ldots, x_{d_2} )\in\mathbb{R}^{d_2} : x_{d_1+1} = \cdots = x_{d_2}=0\}.$$

By making $U_2$ smaller and by replacing $U_2$ with its image under a real analytic diffeomorphism, we can assume that $W_2$ is open in $U_2 \cap \mathbb{R}^{d_1}$ ; indeed, to see that such a change of variables exists, see [Reference Krantz and ParksKP92, Proof of Theorem1.9.5]. Once again, using the uniqueness of analytic continuation of $\Psi_2|_{W_2}$ , we see that $\Psi_2|_{U_2 \cap \mathbb{R}^{d_1}}$ is determined by $\Psi_2|_{W_2}$ . Since the two analytic continuations agree, we see that p belongs to the interior of $L \cap R$ .

Arguing as in the above proof of Theorem 1.2(a), we arrive at the following.

Likewise, for any weight vector $ {\boldsymbol{\omega}} = ({\boldsymbol{\alpha}},{\boldsymbol{\beta}}) \in \mathbb{R}_{>0}^{m+n}$ as in § 1.4, in view of (1.13) the conclusion of the above theorem holds verbatim for the set $\operatorname{UA}_{m,n}(f, {\boldsymbol{\omega}}) $ . Moreover, one can take a subset $\mathcal{W}$ of $\mathbb{R}_{>0}^{m+n}$ and say that $A\in M_{m,n}(\mathbb{R})$ is $(f, \mathcal{W})$ -uniform, denoted with some abuse of notation by $A\in \operatorname{UA}_{m,n}(f, \mathcal{W})$ , if for every sufficiently large t one can find $\textbf{q} \in \mathbb{Z}^n\smallsetminus\{0\}$ and $\textbf{p} \in \mathbb{Z}^m$ such that for any ${{\boldsymbol{\omega}} = ({\boldsymbol{\alpha}},{\boldsymbol{\beta}})}\in \mathcal{W}$ the inequalities (1.12) hold. Clearly this is stronger than being $(f, {\boldsymbol{\omega}})$ -uniform for any ${\boldsymbol{\omega}}\in\mathcal{W}$ . The next statement follows from Theorem 1.5 as easily as the previous one did. It immediately implies both parts of Theorem 1.7.

Theorem 3.4. Let $m,n\in\mathbb{N}$ with $n > 1$ . Suppose that for any $k\in\mathbb{N}$ we are given a non-increasing function $f_k: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ and a subset $\mathcal{W}_k$ of $ \mathbb{R}_{>0}^{m+n}$ such that

(3.3) \begin{equation}{{\sup_{ ({\boldsymbol{\alpha}},{\boldsymbol{\beta}})\in \mathcal{W}_k}\max_{i}\alpha_i <\infty\quad\text{and}\quad \inf_{({\boldsymbol{\alpha}},{\boldsymbol{\beta}})\in \mathcal{W}_k}\min_{j}\beta_j > 0.}}\end{equation}

Then for any countable collection $\{R_\ell \}$ of proper analytic submanifolds of $M_{m,n}(\mathbb{R})$ , the set

$$\bigcap_k \operatorname{UA}_{m,n}(f_k, \mathcal{W}_k) \smallsetminus \bigcup _{\ell} R_\ell$$

is uncountable and dense in $M_{m,n}(\mathbb{R})$ . In particular, the set

$$\bigcap_{{\boldsymbol{\omega}}\in\mathbb{R}_{>0}^{m+n}} \operatorname{UA}_{m,n}(f, {\boldsymbol{\omega}}) \smallsetminus \bigcup _{\ell} R_\ell$$

is uncountable and dense in $M_{m,n}(\mathbb{R})$ for any $\{R_\ell\}$ as above and any non-increasing positive function f.

Proof. Again we take $Y = M_{m,n}(\mathbb{R}) = X_k$ , $ \mathcal{I} = \mathbb{Z}^m\times (\mathbb{Z}^n\smallsetminus\{0\})$ , and $\varphi_k = \operatorname{Id}$ for each k, and, to define the Diophantine systems associated with each $X_k$ , we let

(3.4) \begin{equation} h_{k,(\textbf{p},\textbf{q})} {\, \stackrel{\mathrm{def}}{=}\, } \sup_{{\boldsymbol{\omega}}\in \mathcal{W}_k}\|\textbf{q}\|_{\boldsymbol{\beta}}\quad \text{ and}\quad d_{k,(\textbf{p},\textbf{q})}(A) {\, \stackrel{\mathrm{def}}{=}\, } \inf_{{\boldsymbol{\omega}}\in \mathcal{W}_k}\|A\textbf{q} - \textbf{p}\|_{\boldsymbol{\alpha}}. \end{equation}

Because of (3.3), for any $(\textbf{p},\textbf{q})$ and any k the value $h_{k,(\textbf{p},\textbf{q})}$ is finite and the function $d_{k,(\textbf{p},\textbf{q})}$ is continuous. Thus the same argument, in view of Remark 1.6(2), yields the proof of the first part of the theorem. For the ‘in particular’ part one simply writes $\mathbb{R}_{>0}^{m+n}$ as the union of countably many subsets $\mathcal{W}_k$ satisfying (3.3).

4. Uniformly approximable row vectors on analytic submanifolds and some fractals

Our next goal is to apply Theorem 1.5 to study uniform approximation on submanifolds Y of $M_{m,n}(\mathbb{R})$ . It is clear that the method we are using in this paper (which is essentially Khintchine’s original argument) is not applicable if $\dim Y = 1$ ; indeed, there are very few results in the literature dealing with singular vectors on curves, and many open questions (see e.g. [Reference Poëls and RoyPR22] and the references therein). However, the method does work if $m=1$ , Y is a connected analytic submanifold of $\mathbb{R}^n\cong M_{1,n}(\mathbb{R})$ , and the dimension of Y is at least 2 (see Theorem 1.8).

In this section we will generalize the aforementioned theorem, but first let us observe that the situation gets more complicated when $\min(m,n) > 1$ . The next proposition gives an example of a three-dimensional submanifold (in fact, an affine subspace) of $M_{2,2}(\mathbb{R})$ which does not contain any f-uniform matrices if f decays rapidly enough. Let us recall that a real number $\alpha$ is badly approximable if

(4.1) \begin{equation}{\inf_{q\in \mathbb{Z}\smallsetminus\{0\}} {|q|} \operatorname{dist}(\alpha q, \mathbb{Z})>0.}\end{equation}

Proof. We need to recall the notion of an ordinary Diophantine exponent, which is an asymptotic version of (1.5); namely,

where the distance is computed using the supremum norm on $\mathbb{R}^m$ . It follows from (4.1) that $\omega(\alpha) \leqslant 1$ . Take an arbitrary $\beta\in\mathbb{R}$ such that $1,\alpha,\beta$ are linearly independent over $\mathbb{Q}$ , and consider $\mathbf{v} = \Big(\begin{array}{cc}\alpha \cr\beta\end{array}\Big)$ . Clearly we have

$$\inf_{q\in \mathbb{Z}\smallsetminus\{0\}} q \operatorname{dist}(\mathbf{v} q, \mathbb{Z}^2)>0;$$

hence $\omega(\mathbf{v}) \leqslant 1$ as well.

Now we will use the inequalities due to Jarník relating the ordinary and the uniform exponents of $\mathbf{v}$ and $\mathbf{v}^T$ :

(4.2) \begin{equation}\omega(\mathbf{v}) \geqslant \frac{\hat{\omega}(\mathbf{v})^2}{1-\hat{\omega}(\mathbf{v})}\end{equation}

and

(4.3) \begin{equation}\hat{\omega}(\mathbf{v}) + \frac{1}{\hat{\omega}(\mathbf{v}^T)} = 1\end{equation}

(see [Reference JarníkJar38] and [Reference JarníkJar54] respectively). It follows from (4.2) that $\hat{\omega}(\mathbf{v})\leqslant {(\sqrt{5}-1)}/{2},$ and from (4.3) we can obtain $\hat{\omega}(\mathbf{v}^T) \leqslant {(\sqrt{5}+3)}/{2} = \lambda.$ Thus for any $A\in M_{2,2}(\mathbb{R})$ with $(\alpha,\beta)$ as its row vector we have $\hat{\omega}(A)\leqslant \hat{\omega}(\mathbf{v}^T) \leqslant \lambda.$

Our next result shows that such examples are impossible when $m=1$ and the dimension d of the submanifold Y of $\mathbb{R}^n\cong M_{1,n}$ is at least 2. Furthermore, we will exhibit vectors in Y which are f-uniform of order $d-1$ , as defined in § 1.5.

Theorem 4.2. Let $n\geqslant 2$ , let Y be a d-dimensional connected analytic submanifold of $\mathbb{R}^n$ , where $d\geqslant 2$ , and let $1\leqslant g \leqslant d-1$ . Suppose we are given an arbitrary non-increasing function $f: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ and a countable collection $\{R_\ell \}$ of proper closed analytic submanifolds of Y. Then the set

(4.4) \begin{equation}{Y\cap \operatorname{UA}_{1,n}(f;g) \smallsetminus \bigcup_{\ell\in\mathbb{N}} R_\ell}\end{equation}

is uncountable and dense in Y. In particular, if one in addition assumes that Y is not contained in any proper rational affine subspace of $\mathbb{R}^n$ , then the set $Y\cap \operatorname{UA}^*_{1,n}(f;g) % \smallsetminus \bigcup % _{\ell\in\N} R_\ell $ is uncountable and dense in Y.

To prove Theorem 4.2 we first need to describe the Diophantine system responsible for approximation of order g. Unwrapping Definition 1.9, we can easily see that $\mathbf{x}\in\mathbb{R}^n$ is f-uniform of order g if and only if it is $(\mathcal{X},f)$ -uniform, where $\mathcal{X}$ is the Diophantine system given by

(4.5) \begin{equation}{ \begin{aligned}X = \mathbb{R}^n,\ \ & \mathcal{I} = \mathbb{Z}^g \times \big((\mathbb{Z}^n)^g\smallsetminus\{\text{linearly dependent} \textit{g}-\text{tuples}\}\big),\\ &\qquad h_{p_1, \dots, p_g,\mathbf{q}_1, \ldots, \mathbf{q}_g} = \max_{i=1,\dots,g}\| \mathbf{q}_i \| ,\\ \ &d_{p_1, \dots, p_g,\mathbf{q}_1, \dots, \mathbf{q}_g}(\mathbf{x}) = \max_{i=1,\dots,g} | \mathbf{q}_i \cdot \mathbf{x} -p_i| {.}\quad\end{aligned}}\end{equation}

Note that the zero locus of $d_{p_1, \dots, p_g,\mathbf{q}_1, \dots, \mathbf{q}_g}$ is precisely the intersection of g rational affine hyperplanes $L_{p_i, \textbf{q}_i}$ , which is a codimension g rational affine subspace of $\mathbb{R}^n$ ; moreover, any rational affine subspace of $\mathbb{R}^n$ of codimension g can be (in many different ways) written as such an intersection. Consequently, the collection $\mathcal L$ of all rational affine subspaces of $\mathbb{R}^n$ of codimension g is aligned with $\mathcal{X}$ as in (4.5). It is trivial to check that $\mathcal L$ is totally dense and respects any closed analytic submanifold of $\mathbb{R}^n$ . Hence the case $Y = \mathbb{R}^n$ of Theorem 4.2 easily follows from Theorem 1.5. Let us now prove the general case where Y is an arbitrary d-dimensional analytic submanifold of $\mathbb{R}^n$ . For the proof we will need the following lemma.

Proof. For any $\mathbf{y} \in Y$ one can choose a neighborhood $V\subset \mathbb{R}^n$ containing $\mathbf{y}$ , an open set $U \subset \mathbb{R}^d$ , and a differentiable embedding ${\Psi}: U \to \mathbb{R}^n$ such that ${V} \cap {Y} ={\Psi}({U})$ . Let $\mathbf{x}\in U$ be such that ${\Psi}(\mathbf{x})=\mathbf{y}$ ; then $d_{\mathbf{x}}{\Psi}$ has rank d, and by permuting coordinates in $\mathbb{R}^n$ without loss of generality we can assume that the leftmost $d\times d$ submatrix of $d_{\mathbf{x}} {\Psi}$ is non-singular. Then, in view of the implicit function theorem, we can choose an open subset W of B containing $\mathbf{y}$ such that

$${W\cap Y =\big \{\big(x_1, \dots,x_d, {\psi}(x_1, \dots,x_d)\big) : (x_1, \dots,x_d )\in O\big\},}$$

where O is an open ball in $\mathbb{R}^d$ and $\psi:O \to \mathbb{R}^{n-d}$ is a differentiable function.

Now define

$$\mathcal{L}_{W} {\, \stackrel{\mathrm{def}}{=}\, } \big \{L_M\, \cap\, W\,\cap\, Y : M \text{ is a rational affine subspace of }\mathbb{R}^d\text{ of codimension }g\big\},$$

where

(4.6) \begin{equation}{L_M {\, \stackrel{\mathrm{def}}{=}\, } \big\{(x_1, \dots,x_n): (x_1, \dots,x_d)\in M\big\}.}\end{equation}

Equivalently, $L_M \cap W\cap Y$ can be written as

(4.7) \begin{equation}{\{(x_1, \dots,x_d, \psi(x_1, \dots,x_d)) : (x_1, \dots,x_d )\in M\cap O\}.}\end{equation}

Since $d\geqslant 2$ and $1\leqslant g \leqslant d-1$ , the collection

$$\{M \cap O : M \text{ is a rational affine subspace of }\mathbb{R}^d\text{ of codimension g}\}$$

is totally dense in O, which immediately implies that $\mathcal{L}_{W}$ is totally dense in $Y\cap W$ .

Let us now show that the same method can produce uniformly approximable vectors on certain fractals. Below is an example from [Reference Kleinbock, Moshchevitin and WeissKMW21].

Theorem 4.4. Let $n \geqslant 2$ and let $Y_1, \ldots, Y_n$ be perfect subsets of $\mathbb{R}$ such that

(4.8) \begin{equation}{\mathbb{Q} \cap Y_k \text{is dense in }Y_k\text{ for each }k \in \{1,2\}.}\end{equation}

Let $Y = \prod_{j=1}^n Y_j$ . Suppose we are given an arbitrary non-increasing function $f: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ and a countable collection $\{R_\ell \}$ of proper closed analytic submanifolds of $\mathbb{R}^n$ . Then

$$Y\cap \operatorname{UA}_{1,n}(f) \smallsetminus \bigcup_{\ell\in\mathbb{N}} R_\ell$$

is uncountable and dense in Y. In particular, $Y\cap \operatorname{UA}^*_{1,n}(f%;g ) % \smallsetminus \bigcup % _{\ell\in\N} R_\ell $ is uncountable and dense in Y.

Proof. Let $\mathbf{e}_1, \ldots, \mathbf{e}_n$ be the standard basis vectors in $\mathbb{R}^n$ , and let $\{A_i\} $ be the collection of all rational affine hyperplanes of $\mathbb{R}^n$ which are normal to one of $\mathbf{e}_1$ and $\mathbf{e}_2$ and have non-trivial intersection with Y; that is, each of the $A_i$ is of the form

$${A_i = \left\{ \mathbf{x} \in \mathbb{R}^{n} : x_{k_i} = r_i \right\},\text{ where } r_i \in \mathbb{Q} \text{ and } k_i \in \{1,2\};}$$

note that necessarily we have $r_i\in {Y_{k_i}}$ . We claim that the collection $\{ Y \cap A_i\}$ , which is obviously aligned with $\mathcal{X}_{1,n}$ , is totally dense. Indeed, (1.7) clearly follows from (4.8). To prove (1.9), take an open subset W of $\mathbb{R}^n$ of the form $I_1\times\cdots\times I_n$ , where $I_j$ are open intervals in $\mathbb{R}$ , and suppose that

(4.9) \begin{equation}{A = \{ \mathbf{x} \in \mathbb{R}^{n} : x_1 = r\}}\end{equation}

intersects with Y non-trivially, that is, we have $r\in Y_1$ . Then, again in view of (4.8), the union of subspaces $A_j$ such that $A_j\cap A \cap Y\cap W \ne\varnothing$ is dense in $(I_1\cap Y_1)\times\mathbb{R}\times \cdots\times \mathbb{R}$ , and hence the union of corresponding intersections $Y \cap A_j$ is dense in $Y\cap(I_1\times\mathbb{R}\times \cdots\times \mathbb{R})$ .

It remains to prove that the collection $\{ Y \cap A_i\}$ respects an arbitrary closed analytic submanifold R of $\mathbb{R}^n$ . Suppose that, for A as in (4.9), the intersection $Y\cap A \cap R$ has a non-empty interior in $Y\cap A$ . Take a point $\mathbf{x} = (x_1,\dots,x_n)$ in the interior of $Y\cap A \cap R$ ; then there exists a neighborhood W of $\mathbf{x}$ in $\mathbb{R}^n$ such that $Y\cap A \cap R$ contains $\big(\{x_1\} \times Y_2 \times \cdots \times Y_n\big)\cap W$ . Since each of the sets $Y_i$ is perfect, there are sequences in $Y\cap A \cap R$ approaching $\mathbf{x}$ from each coordinate direction. This makes it possible to determine all partial derivatives of all orders of any analytic function on A at $(x_1,\dots,x_n)$ by its values on $Y\cap A \cap R$ . In particular, since R is a closed analytic manifold, it has to contain A, and hence it is respected by $Y\cap A$ .

Now we can conclude that Theorem 1.5 applies and implies the density of $Y\cap \operatorname{UA}_{1,n}(f) \smallsetminus \bigcup_{\ell\in\mathbb{N}} R_\ell$ for any countable set of proper closed analytic submanifolds $R_\ell$ of $\mathbb{R}^n$ . The uncountability follows as well, since the sets $\{ Y \cap A_i\}$ have no isolated points.

Remark 4.5. Our arguments can be adapted to many other fractal sets. We sketch one more example involving a ‘rational Koch snowflake’ which can be treated by our method. To define it, let $\alpha, \beta, \gamma, \delta$ be positive rational numbers, where $\alpha < \beta < \delta < 1$ , and let Y be the attractor of the iterated function system $\{\varphi_1,\varphi_2, \varphi_3, \varphi_4\}$ , where $\varphi_i : \mathbb{R}^2 \to \mathbb{R}^2$ is the unique orientation-preserving contracting similarity map sending the points $\mathbf{x}_0 = (0,0)$ and $\mathbf{y}_0 = (1,0)$ , to the points $\mathbf{x}_i, \mathbf{y}_i$ defined by

\[\begin{split} \mathbf{x}_1 = (0,0), \ \ \ &\mathbf{y}_1 = ( \alpha, 0) \\ \mathbf{x}_2 = \mathbf{y}_1,{\kern.25pt} \ \ \ \ \ \ &\mathbf{y}_2 = ( \beta, \gamma) \\ \mathbf{x}_3 = \mathbf{y}_2,{\kern.25pt} \ \ \ \ \ \ &\mathbf{y}_3 = (\delta, 0) \\ \mathbf{x}_4 = \mathbf{y}_3,{\kern.25pt} \ \ \ \ \ \ &\mathbf{y}_4 = ( 1, 0). \end{split}\]

Then Y is a topological curve (continuous image of a segment) of Hausdorff dimension greater than one. The usual version of the Koch snowflake has a similar description, with $\alpha = \frac13,\,\beta = \frac12,\, \gamma = \frac{\sqrt{3}}{6}, \delta = \frac23$ (see [Reference FalconerFal04, Figure 0.2]).

For our application, it is important to note that if $\theta$ is the slope of the line from $\mathbf{x}_2$ to $\mathbf{y}_2$ , then Y contains a dense collection of rational points $Y_0$ , and the horizontal lines and lines of slope $\theta$ through points of $Y_0$ intersect Y in an uncountable perfect set. This property makes it possible to repeat the proof of Theorem 5.2, taking the sets $\{A_i\}$ to be the lines having these two slopes and passing through $Y_0$ .

5. Vectors that are k-singular for all k

In this section we prove a strengthening of Theorem 1.11. To state it, let us consider a definition generalizing the notion of k-singular vectors. Namely, for a non-increasing $f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ we say that $\mathbf{x}\in\mathbb{R}^n$ is f-uniform of degree k if for any sufficiently large t there exists a polynomial $P \in \mathbb{Z}[X_1, \ldots, X_n]$ such that $\deg (P) \leqslant k, \ H(P) \leqslant t,$ and $|P(\mathbf{x})| \leqslant f(t)$ . Clearly $\mathbf{x}$ , viewed as a row vector, is k-singular (as defined in § 1.6) if and only if it is $\varepsilon\phi_{N_k}$ -uniform of degree k, where $N_k$ is as in (1.14). Also it is clear that k-algebraic vectors (again see § 1.6 for a definition) are f-uniform of degree k for any positive f.

Let us say that a submanifold ${Y} \subset {\mathbb{R}^n}$ is transcendental if Y is not contained in an algebraic subvariety defined over $\mathbb{Q}$ or, more concretely, if there is no nonzero polynomial in $\mathbb{Z}[X_1, \ldots, X_n]$ which vanishes on Y.

This result extends [Reference Kleinbock, Moshchevitin and WeissKMW21, Theorem 1.7]; namely, there it was shown that Y as above contains uncountably many singular vectors.

The proof is based on the following observation. For $\mathbf{x} \in \mathbb{R}^n$ , denote by $\varphi_k(\mathbf{x})$ the vector with coordinates consisting of all non-constant monomials of degree at most k in n variables $x_1,\dots,x_n$ , viewed as a row vector in $\mathbb{R}^{N(k,n)}$ , and for each k consider the standard Diophantine system $\mathcal{X}_{1,N(k,n)}$ on $X_k = \mathbb{R}^{N(k,n)}\cong M_{1,N(k,n)}$ with distance functions from rational affine hyperplanes and standard heights. Then it is clear that $\mathbf{x} \in \mathbb{R}^n$ is $f_k$ -uniform of degree k if and only if $\varphi_k(\mathbf{x}) \in \operatorname{UA}_{1,N(k,n)}(f_k)$ . The proof is as follows.

• • a neighborhood W of $\mathbf{y}$ such that $Y\cap W$ is a graph of an analytic function $\mathbb{R}^d\to \mathbb{R}^{n-d}$ ; and

• • (using Lemma 4.3 with $g=1$ ) a totally dense collection $\mathcal{L}_W$ of intersections of some rational affine hyperplanes of $\mathbb{R}^n$ with $Y\cap W$ .

It is clear that $\mathcal{L}_W$ as above is aligned with $\mathcal{X}_{1,N(k,n)}$ via $\varphi_k$ : indeed, if $\pi: \mathbb{R}^{N(k,n)} \to \mathbb{R}^n$ denotes the projection onto the first n coordinates and L is a rational affine hyperplane in $\mathbb{R}^n$ , then $\varphi_k(L)$ is contained in $\pi^{-1}(L)$ , which is a rational affine hyperplane in $ \mathbb{R}^{N(k,n)}$ . Now let $\{R_\ell\}$ be the collection of sets of the form $\{\mathbf{y} \in Y: P(\mathbf{y}) =0\}$ , where P ranges over all nonzero polynomials in $\mathbb{Z}[X_1, \ldots, X_n]$ . Since Y is assumed to be transcendental, it follows that each of the sets $R_\ell$ is a proper subset of Y. It is easy to show that $\mathcal{L}_W$ respects $R_\ell$ : indeed, any element L of $\mathcal{L}_W$ is of the form (4.7) with analytic ${\psi}$ ; furthermore, the fact that $L\cap R_\ell$ has non-empty interior in L implies that $P\big(x_1, \dots,x_d, \psi(x_1, \dots,x_d)\big) = 0$ on an open subset of an affine hyperplane M of $\mathbb{R}^d$ . Since $d\geqslant 2$ and ${\psi}$ is analytic, it follows that $L\subset R_\ell$ . Hence Theorem 5 applies, and we conclude that the set

$${\bigcap_{k} \varphi_k^{-1}\big(\operatorname{UA}_{1,N(k,n)}(f_k)\big) \smallsetminus \bigcup_{{{\ell}}} R_{{\ell}}},$$

which contains the set of vectors that are not algebraic and are $f_k$ -uniform of degree k for all $k \in \mathbb{N}$ , is uncountable and dense in Y for any choice of non-increasing functions $f_k: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ .

Using similar ideas and arguing as in the proof of Theorem 4.4 and Remark 4.5, one can establish a similar result for fractals.

6. Sumsets of sets of totally irrational uniformly approximable vectors

Our goal in this section is to prove Theorem 1.12, which states that when $n\geqslant 3$ , the sum of $\operatorname{UA}^*_{1,n}(f_1) $ and $\operatorname{UA}^*_{1,n}(f_2) $ is $\mathbb{R}^n$ for any pair of approximating functions $f_1,f_2$ . Clearly, by replacing each of the functions $f_i$ with $\min(f_1,f_2)$ , it is enough to take a non-increasing $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ and prove that $\operatorname{UA}^*_{1,n}(f) +\operatorname{UA}^*_{1,n}(f) =\mathbb{R}^n$ . Equivalently, since $\operatorname{UA}^*_{1,n}(f) $ coincides with $ -\operatorname{UA}^*_{1,n}(f)$ , we need to show that for any fixed $\mathbf{z} \in \mathbb{R}^n$ the intersection of $\operatorname{UA}^* _{1,n}(f) $ and its translate $\operatorname{UA}^*_{1,n}(f) - \mathbf{z}$ is not empty. Unwrapping the definitions, we see that this amounts to finding $\mathbf{x}\in\mathbb{R}^n$ such that both $\mathbf{x}$ and $\mathbf{x} +\mathbf{z}$ are totally irrational and such that for every sufficiently large t one can find $\textbf{q} \in \mathbb{Z}^n\smallsetminus\{0\}$ and $p \in \mathbb{Z}$ with

$$|\mathbf{x}\cdot\textbf{q} -p|\leqslant f(t) \quad \mathrm{and}\quad \|\textbf{q}\|\leqslant t,$$

and also $\textbf{q}' \in \mathbb{Z}^n\smallsetminus\{0\}$ and $p' \in \mathbb{Z}$ with

$$|(\mathbf{x}-\mathbf{z})\cdot\textbf{q}' -p'|\leqslant f(t) \quad \mathrm{and}\quad\|\textbf{q}'\|\leqslant t.$$

Rewriting the two systems of inequalities above as

$$\max(|\mathbf{x}\cdot\textbf{q} -p|,|(\mathbf{x}-\mathbf{z})\cdot\textbf{q}' -p'|) \leqslant f(t) \quad \mathrm{and}\quad \max(\|\textbf{q}\|,\|\textbf{q}'\|)\leqslant t,$$

we see that the problem reduces to a new Diophantine system

$${\mathcal{X}_\mathbf{z} := \big( \mathbb{R}^n, \{d_{p,p',\textbf{q},\textbf{q}'}\}, \{h_{p,p',\textbf{q},\textbf{q}'}\}\big)}$$

where $d_{p,p',\textbf{q},\textbf{q}'}(\mathbf{x}) := \max(|\mathbf{x}\cdot\textbf{q} -p|,|(\mathbf{x}-\mathbf{z})\cdot\textbf{q}' -p'|)$ , $h_{p,p',\textbf{q},\textbf{q}'} := \max(\|\textbf{q}\|,\|\textbf{q}'\|)$ , and $(p,p', \textbf{q},\textbf{q}')$ runs through $\mathbb{Z}\times \mathbb{Z}\times (\mathbb{Z}^n\smallsetminus\{0\})\times(\mathbb{Z}^n\smallsetminus\{0\})$ . Note that the zero locus of $d_{p,p',\textbf{q},\textbf{q}'}$ is precisely the intersection $L_{p, \textbf{q}} \cap (L_{p', \textbf{q}'} + \mathbf{z})$ of two hyperplanes in $\mathbb{R}^n$ ; in other words, the collection

\begin{equation*}{\mathcal{L}_\mathbf{z}{\, \stackrel{\mathrm{def}}{=}\, } \bigg\{L_{p, \textbf{q}} \cap (L_{p', \textbf{q}'} + \mathbf{z})\, \bigg|\, \begin{aligned}\ p,p'\in\mathbb{Z},\ \textbf{q},\textbf{q}'\in\mathbb{Z}^n\smallsetminus\{0\},\\ \textbf{q} \text{ and } \textbf{q}' \text{ are not proportional}\end{aligned}\bigg\}}\end{equation*}

is aligned with the Diophantine system ${\mathcal{X}_\mathbf{z}}$ . The crucial step of the proof of Theorem 1.12 will be the following lemma.

Note that we are not able to prove the stronger form of total density, that is, the validity of (1.9) for $\mathcal{L} = \mathcal{L}_\mathbf{z}$ .

Sublemma 6.2. Let $n \geqslant 3$ and let N be an affine hyperplane of $\mathbb{R}^n$ . Then the collection

$$\mathcal{L}(N){\, \stackrel{\mathrm{def}}{=}\, } \{M\cap N: M \text{ is a rational affine hyperplane of }\mathbb{R}^n, M\ne N\}$$

is totally dense in N; furthermore, the union in (1.9) is dense in N.

Proof. By permuting coordinates, without loss of generality we can express N in the form

$$x_n = a_1x_1 + \cdots+a_{n-1}x_{n-1} + a_n\quad\text{where }a_i\in\mathbb{R};$$

in this way the projection $\pi:(x_1,\dots,x_n)\mapsto(x_1,\dots,x_{n-1})$ maps N bijectively onto $\mathbb{R}^{n-1}$ . Furthermore, any $L\in \mathcal{L}(N)$ is of the form

(6.1) \begin{equation}{\{\mathbf{x}\in\mathbb{R}^n: x_n = a_1x_1 + \cdots+a_{n-1}x_{n-1} + a_n, \ q_1x_1 + \cdots+q_{n}x_{n} = p\}}\end{equation}

for some $\mathbf{q}\in\mathbb{Z}^n\smallsetminus\{0\}$ and $p\in\mathbb{Z}$ . It suffices to prove that the collection $\pi\big(\mathcal{L}(N)\big)$ of affine hyperplanes of $\mathbb{R}^{n-1}$ is totally dense in $\mathbb{R}^{n-1}$ . By considering the case $q_n = 0$ in (6.1), it is easy to see that $\pi\big(\mathcal{L}(N)\big)$ contains the collection of all rational affine hyperplanes of $\mathbb{R}^{n-1}$ . The latter collection, in additional to being totally dense, has the following property: for any affine hyperplane L of $\mathbb{R}^{n-1}$ , any open $W\subset \mathbb{R}^{n-1}$ with $W\cap L \ne\varnothing$ and any $\mathbf{y}\in\mathbb{Q}^{n-1}\smallsetminus L$ there exists $\mathbf{y}'\in\mathbb{Q}^{n-1}$ such that the (rational) line passing through $\mathbf{y}$ and $\mathbf{y}'$ intersects $W\cap L$ . This clearly implies that the union of all rational affine hyperplanes touching $W\cap L \ne\varnothing$ is dense in $\mathbb{R}^{n-1}$ , and hence the total density of $\pi\big(\mathcal{L}(N)\big)$ .

The above sublemma implies, in particular, that for any $N = L_{p', \textbf{q}'} + \mathbf{z}$ the union

$$\bigcup_{p\in\mathbb{Z},\,\textbf{q}\in\mathbb{Z}^n\setminus\mathbb{R}\textbf{q}} L_{p, \textbf{q}}\cap N$$

is dense in N. Since the union of all rational affine hyperplanes translated by $\mathbf{z}$ is dense in $\mathbb{R}^n$ , it follows that $\bigcup_{L\in{\mathcal L}_\mathbf{z}}L$ is dense in $\mathbb{R}^n$ , i.e. condition (1.7) holds.

Now take $L\in{\mathcal L}_\mathbf{z}$ , that is, $L = L_{p, \textbf{q}} \cap (L_{p', \textbf{q}'} + \mathbf{z}) $ such that $ \textbf{q}$ and $\textbf{q}'$ are not proportional, and choose an open subset W of $\mathbb{R}^n$ with $L\cap W\ne\varnothing$ . For brevity we write $M {\, \stackrel{\mathrm{def}}{=}\, } L_{p, \textbf{q}}$ and ${N} {\, \stackrel{\mathrm{def}}{=}\, } L_{p', \textbf{q}'} + \mathbf{z}$ . It follows from Sublemma 6.2 that the union of $P\cap N$ over all rational affine hyperplanes $P\ne N$ that satisfy $P\cap W\cap L\ne\varnothing$ is dense in N. Applying a translation by $-\mathbf{z}$ to the above conclusion one gets that the union of $(P+\mathbf{z})\cap M$ over all rational affine hyperplanes P with $P+\mathbf{z} \ne M$ and $P\cap W\cap L\ne\varnothing$ is dense in M. It follows that the closure of the union of all $L'\in{\mathcal L}_\mathbf{z}$ such that $L'\cap W\cap L\ne\varnothing$ contains $M\cup N$ , and thus it cannot be a subset of a single affine hyperplane.

Now, since $\mathcal{L}_\mathbf{z}$ obviously respects any affine subspace in $\mathbb{R}^n$ , we can apply Theorem 1.5 to $\mathcal X_\mathbf{z}$ with $\mathcal{R} = \{R_\ell\}$ being an arbitrary countable collection of affine hyperplanes of $\mathbb{R}^n$ . This immediately yields the proof of Theorem 1.12 in a slightly more general form, with $\operatorname{UA}^*_{1,n}(f_k)$ , $k=1,2$ , in (1.15) replaced by $\operatorname{UA}_{1,n}(f_k) \smallsetminus\bigcup_\ell R_\ell$ for $\mathcal{R}$ as above.

Remark 6.3. It is not hard to show, by adapting the above proof, that for any $n \geqslant 3$ , any positive non-increasing $f_1, \ldots, f_{n-1}$ , and any $\mathbf{z}_1,\ldots, \mathbf{z}_{n-1}$ , we have

$$\bigcap_{k=1}^{n-1}( \operatorname{UA}^*_{1,n}(f_k) - \mathbf{z}_k) \neq \varnothing.$$

Note that our method gives no information on the intersection of translates of $\operatorname{UA}^*_{1,2}(f)$ . In particular, it is an open problem to determine whether for any non-increasing $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ the sumset of $\operatorname{UA}^*_{1,2}(f)$ with itself coincides with $\mathbb{R}^2$ .

7. Transference and improved rates for vectors in analytic submanifolds of $\mathbb{R}^n$

Given $\mathbf{x} = (x_1, \ldots, x_n)\in\mathbb{R}^n$ and real numbers $\tau, \varepsilon, t, \eta$ , define the following parallelepipeds:

$$\Pi^{\tau,\varepsilon} {\, \stackrel{\mathrm{def}}{=}\, } \Big\{(z_0,z_1,\ldots,z_n) \in \mathbb{R}^{n+1}:\,\,\max_{1\leqslant j \leqslant n} |z_j|\leqslant \tau,\, |z_0 +z_1x_1+\cdots+z_n x_n| \leqslant \varepsilon\Big\}$$

and

$$\Pi_{t, \eta} {\, \stackrel{\mathrm{def}}{=}\, }\Big\{(z_0,z_1,\ldots,z_n) \in \mathbb{R}^{n+1}:\,\,|z_0|\leqslant t,\,\max_{1\leqslant j \leqslant n} |z_0x_j - z_j|\leqslant \eta\Big\}.$$

Note that $\Pi^{\tau, \varepsilon}$ encodes information about approximations to $\mathbf{x}$ as a linear form, while $\Pi_{t, \eta}$ encodes approximations to $\mathbf{x}$ as a vector.

With these definitions, we have the following standard transference result:

Lemma 7.1. Let $1\leqslant g \leqslant n$ , suppose that $\tau, \varepsilon, t, \eta$ satisfy

(7.1) \begin{equation} \frac{\eta}{t} = \frac{\varepsilon}{\tau} \end{equation}

and

(7.2) \begin{equation}\eta^{n-g}t = {(4n)^{2n}} \tau^{g},\end{equation}

and assume that the parallelepiped $\Pi^{\tau,\varepsilon} $ contains g linearly independent integer points. Then $\Pi_{t, \eta}$ contains a nonzero integer point.

Proof. Let $ \mathbf{z}_1, \ldots ,\mathbf{z}_g$ be linearly independent integer points in $\Pi^{\tau, \varepsilon}$ , let

$${L} {\, \stackrel{\mathrm{def}}{=}\, } {\rm span}\, (\mathbf{z}_1,\ldots,\mathbf{z}_g)$$

be the space generated by these points, and denote the orthogonal complement of L by ${L}^\perp$ . Clearly $\Lambda {\, \stackrel{\mathrm{def}}{=}\, } {L}\cap\mathbb{Z}^{n+1}$ is a lattice in L, and since ${L}^\perp$ is rational, $\Lambda^\perp = {L}^\perp\cap\mathbb{Z}^{n+1}$ is a lattice in ${L}^\perp.$ It is well known that the covolumes $\mathrm{covol}({L}/\Lambda)$ and $\mathrm{covol}({L}^\perp/\Lambda^\perp)$ are equal to each other. Now define $ \Omega {\, \stackrel{\mathrm{def}}{=}\, } {L} \cap \Pi^{\tau, \varepsilon}$ and $ \Omega^\perp {\, \stackrel{\mathrm{def}}{=}\, } {L}^\perp \cap \Pi_{t, \eta}$ , and denote by ${\rm vol}_g \Omega$ and ${\rm vol}_{n+1-g} \Omega^\perp$ , respectively, the g-dimensional and $(n-g+1)$ -dimensional volumes of $\Omega$ and $\Omega^\perp$ .

Denote the vector $(1,x_1,\dots,x_n)\in\mathbb{R}^{n+1}$ by $\mathbf{x}_0$ , and let

$${M} =\big\{(z_0,z_1,\ldots,z_n) \in \mathbb{R}^{n+1}:\,z_0 +z_1x_1+\cdots +z_nx_n = 0\big\}$$

be the hyperplane orthogonal to $\mathbf{x}_0$ . Let us define

$$\theta_{\max}{\, \stackrel{\mathrm{def}}{=}\, } \max_{\mathbf{u} \in {L},\, \mathbf{v}\in {M}} \,(\text{angle between}\,\, \mathbf{u}\,\,\text{and}\,\, \mathbf{v})$$

to be the maximal angle between vectors in L and M, and

$$\theta_{\min} = \min_{\mathbf{u} \in {L}^\perp} (\text{angle between}\,\, \mathbf{u}\,\,\text{and}\,\, \mathbf{x}_0).$$

It is easy to see that $\theta_{\min}\leqslant \theta_{\max}$ . Indeed, let $\mathbf{u}_0$ be the minimizer in the definition of $\theta_{\min}$ , let V be the two-dimensional subspace generated by $\mathbf{x}_0$ and $\mathbf{u}_0$ , and let $\mathbf{u}_0^\perp$ and $\mathbf{x}_0^\perp$ be unit vectors in V perpendicular to $\mathbf{u}_0$ and $\mathbf{x}_0$ respectively. Then

$$\theta_{\min} = \text{ angle } (\mathbf{u}_0, \mathbf{x}_0) = \text{ angle } (\mathbf{u}_0^\perp, \mathbf{x}_0^\perp) \leqslant \theta_{\max}.$$

Let us now consider two cases. Suppose that $\sin \theta_{\min} \leqslant \frac{\eta}{t}$ . Then we have a lower bound

(7.3) \begin{equation} {\rm vol}_{n-g+1} \Omega^\perp \geqslant {t \eta^{n-g}} ,\end{equation}

and an upper bound

(7.4) \begin{equation} {\rm vol}_g \Omega \leqslant 2{n}^{g/2} \tau^g .\end{equation}

Note that $\Omega$ contains the convex hull of $\{\pm \mathbf{z}_1,\ldots, \pm \mathbf{z}_g\}$ , and the parallelepiped

$$\bigg\{\mathbf{z} = \lambda_1 \mathbf{z}_1+ \cdots+\lambda_g\mathbf{z}_g,\,\, \max_{1\leqslant i\leqslant g}|\lambda_i|\leqslant \frac{1}{2}\bigg\}\subset \frac{g}{2} \cdot \Omega$$

contains a fundamental domain for $\Lambda$ . Therefore we have

By the Minkowski Convex Body Theorem there exists a nonzero integer point in $\Omega^\perp \subset \Pi_{t, \eta}$ . Hence we are done in this case.

Now suppose that $\sin \theta_{\min} > {\eta}/{t}$ . Then

(7.5) \begin{equation}{\rm vol}_{n-g+1} \Omega^\perp \geqslant \frac{\eta^{n-g+1}}{\sin \theta_{\min}}\end{equation}

and

(7.6) \begin{equation} {\rm vol}_g \Omega \leqslant {n}^{g/2} \tau^{g-1} \cdot \frac{\varepsilon}{\sin \theta_{\max}}.\end{equation}

Arguing as in the previous case but using (7.5) and (7.6) instead of (7.3) and (7.4), we obtain

and, again by the Minkowski Convex Body Theorem, we get a nonzero integer point in $\Omega^\perp \subset \Pi_{t, \eta}$ .

Now let us state and prove a generalization of Theorem 1.13.

Theorem 7.2. Let $n\geqslant 2$ , let Y be a d-dimensional connected analytic submanifold of $\mathbb{R}^n\cong M_{n,1}(\mathbb{R})$ , where $d\geqslant 2$ , and let $\{R_\ell \}$ be a countable collection of proper closed analytic submanifolds of Y. Suppose that a continuous non-decreasing function f satisfies (1.17). Then the set

$${Y\cap \operatorname{UA}_{n,1}(f) \smallsetminus \bigcup_{\ell\in\mathbb{N}} R_\ell}$$

is uncountable and dense in Y-2pt. In particular, if one in addition assumes that Y is not contained in any proper rational affine subspace of $\mathbb{R}^n$ , then the set $Y\cap \operatorname{UA}^*_{n,1}(f) % \smallsetminus \bigcup % _{\ell\in\N} R_\ell $ is uncountable and dense in Y-2pt.

Proof. For $t> 0$ and $g {\, \stackrel{\mathrm{def}}{=}\, } d-1$ , let us set $\eta {\, \stackrel{\mathrm{def}}{=}\, } f (t)$ and $\tau {\, \stackrel{\mathrm{def}}{=}\, } ({(\eta^{n-g}t)}/{ {(4n)^{2n}}})^{1/g}, $ so that (7.2) is satisfied, and the function $t\mapsto \tau$ is continuous. By (1.17), we have that $ \tau \to \infty$ monotonically when $t\to \infty$ . The map $t \mapsto \tau$ is thus bijective, and hence one can consider the inverse map $ \tau\mapsto t(\tau)$ and define a positive function

$$\varepsilon = {h}(\tau) {\, \stackrel{\mathrm{def}}{=}\, } \frac{\tau f \big(t(\tau)\big)}{t(\tau)},$$

so that (7.1) holds. Note that h is non-increasing, since so is the function

$$h\big(\tau(t)\big) = \frac{( {f(t)^{n-g}t}/{ {(4n)^{2n}}} )^{1/g}f(t)}t = \frac{f(t)^{n/g}}{ {(4n)}^{2n/g}t^{1-1/g}}.$$

We will prove the implication

(7.7)

This, in view of Theorem 4.2, will immediately imply the conclusion of Theorem 7.2.

To prove (7.7), note that $\mathbf{x}^T\in \operatorname{UA}_{1,n}(h;g)$ amounts to saying that for any large enough $\tau$ the parallelepiped $\Pi^{\tau,\varepsilon}$ contains g linearly independent integer points. In view of Lemma 7.1 we get that for all t large enough, the parallelepiped $\Pi_{t, \eta}$ contains a nonzero integer point $\mathbf{z} =(q, \mathbf{p})$ . For t large enough, the first coordinate q will be nonzero; thus we can find $q \in \mathbb{Z}\smallsetminus\{0\}$ and $\mathbf{p} \in \mathbb{Z}^n$ such that $q \leqslant t$ and $\|q\mathbf{x} -\mathbf{ p} \| \leqslant f(t).$

8. Further applications

Most of the results described in this section are not proved in this paper; the proofs will appear elsewhere.

8.1 Irrationality measure functions and the Kan–Moshchevitin phenomenon

Another convenient way to describe various results in the theory of Diophantine approximation is through irrationality measure functions. Namely, given $A\in M_{m,n}(\mathbb{R})$ , one defines its irrationality measure function by

$$\psi_{A}: \mathbb{R}_{>0} \to \mathbb{R}_{>0}, \quad \psi_{A}(t) {\, \stackrel{\mathrm{def}}{=}\, }\inf \big\{\|A\mathbf{q} - \textbf{p}\| : \mathbf{q} \in \mathbb{Z}^n\smallsetminus\{0\}, \ \|\mathbf{q} \| \leqslant t, \ \textbf{p} \in \mathbb{Z}^m \big\}.$$

Then it is easy to see that A is f-uniformFootnote ⁴ if and only if $\psi_{A}(t) \leqslant f(t)$ for all large enough t. Similarly, for an arbitrary Diophantine system $\mathcal{X} = ( X, \mathcal{D}, \mathcal{H})$ one can introduce the irrationality measure function associated to $x\in X$ :

$$\psi_x(t) {\, \stackrel{\mathrm{def}}{=}\, } \inf \{ d_s(x): h_s\leqslant t\}.$$

Then $x\in \operatorname{UA}_{\mathcal{X}}(f)$ if and only if $\psi_x(t) \leqslant f(t)$ for all large enough t.

In 2009 the following result was proved by Kan and Moshchevitin [Reference Kan and MoshchevitinKM10]: in the standard Diophantine system corresponding to approximations to one real number ( $m=n=1$ ), for any two different real numbers x,y with

$$\psi_x(t) >0,\quad\psi_y(t) >0 \quad \forall \, t$$

(a condition equivalent to $x,y\notin\mathbb{Q}$ ), the difference $\psi_x(t) -\psi_y(t)$ changes its sign infinitely often as $t \to \infty$ . This phenomenon is not present when $\max(m,n) > 1$ . More generally, as long as the assumptions of our main theorem (Theorem 1.5) are satisfied, there exist pairs of points whose irrationality measure functions do not exhibit the pattern of infinitely many changes of sign. More precisely, suppose a Diophantine system

$${\mathcal{X} = \big( X, \mathcal{D} = \{d_{s}: s\in\mathcal{I}\}, \mathcal{H} = \{h_{s}: s\in\mathcal{I}\}\big)}$$

and a countable collection $\mathcal{L}$ of closed subsets of X satisfy the assumptions of Theorem 1.5 with $Y = X$ , $\varphi_k\equiv \operatorname{Id}$ and $\mathcal{R} = \{{d_s}^{-1}(0) : s\in \mathcal{I}\}$ . Then for any $x\in X$ such that

$$d_{s}(x)\ne 0\quad\Longleftrightarrow \quad \psi_x(t) >0 \,\,\,\forall \, t$$

one can apply the theorem with $f = \psi_x$ and conclude that there exists a dense set of $y\in X$ such that $\psi_y(t) >0$ for all t and $\psi_y(t)< \psi_x(t)$ for all large enough t.

8.2 Inhomogeneous approximation and approximation with restrictions

It is natural to consider the problems of uniform approximation for systems of linear forms in the inhomogeneous set-up; that is, fix $\mathbf{b}\in\mathbb{R}^m$ and, instead of (1.2), look for non-trivial integer solutions of the system

(8.1) \begin{equation}{{\|A\textbf{q} + \mathbf{b} - \textbf{p} \|\leqslant f(t)} \quad \mathrm{and}\quad \|\textbf{q}\|\leqslant t.}\end{equation}

This calls for the consideration of the collection

(8.2) \begin{equation}{\begin{aligned}\{L_{\textbf{p},\textbf{q},\mathbf{b}} : \textbf{q} \in\mathbb{Z}^n\smallsetminus\{0\}, \ \textbf{p}\in\mathbb{Z}^m\}\quad \text{where }L_{\textbf{p},\textbf{q},\mathbf{b}} {\, \stackrel{\mathrm{def}}{=}\, } \{A\in M_{m,n}(\mathbb{R}): A\textbf{q} + \mathbf{b}= &\textbf{p}\},\end{aligned}}\end{equation}

and the proof of its total density for arbitrary $\mathbf{b}\in\mathbb{R}^n$ . And indeed it turns out to be possible to achieve this when $n > 1$ , and thereby construct f-uniform systems of affine forms with an arbitrary fixed translation part. Moreover, in a forthcoming joint work with Hong and Neckrasov we take subsets P of $\mathbb{R}^m$ and Q of $\mathbb{Z}^n$ , and say that $A\in M_{m,n}(\mathbb{R})$ is f-uniform with respect to (P,Q) if for all large enough $t>0$ there exist $\textbf{q} \in Q$ and $\textbf{p} \in P$ satisfying (1.2). The classical theory of homogeneous approximation corresponds to $P = \mathbb{Z}^m$ and $Q = \mathbb{Z}^{{n}}\smallsetminus\{0\}$ . The following theorem can be proved (work in progress).

Theorem 8.1. The collection ${{\{L_{\textbf{p},\textbf{q}}}: \textbf{p}\in P,\,\textbf{q}\in Q\}}$ of subspaces of $M_{m,n}(\mathbb{R})$ is totally dense if:

1. (a) $Q = \mathbb{Z}^n\smallsetminus\{0\}$ and P is a subgroup of $\mathbb{Z}^m$ of rank $>m-n+1$ (restricted numerators);

2. (b) $Q = Q_1\times\cdots\times Q_n\subset \mathbb{Z}\times\cdots\times\mathbb{Z}$ , where at least two of the sets $Q_i$ are infinite (restricted denominators), and P is of bounded Hausdorff distance from $\mathbb{R}^m$ .

Consequently (modulo Theorem 1.5) for any non-increasing $f: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ the set of $m \times n$ matrices that are f-uniform with respect to (P,Q) and not contained in a given countable family of proper analytic submanifolds of $M_{m,n}(\mathbb{R})$ is uncountable and dense.

Note that both (a) and (b) implicitly assume that $n > 1$ , and the set-up of (b) includes inhomogeneous approximation by letting $P = \mathbb{Z}^m+\mathbf{b}$ for a fixed $\mathbf{b}\in\mathbb{R}^m$ . When $n=1$ , it is clear that for any fixed $\mathbf{b}\in\mathbb{R}^m$ the collection (8.2) is not totally dense. On the other hand one can study a doubly metric version of the problem, that is, the set of pairs $(A,\mathbf{b}$ ) such that (8.1) has a non-trivial integer solution for large enough t. It was recently documented by the second named author [Reference MoshchevitinMos24] that for any $m\in\mathbb{N}$ and any non-increasing $f: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ there exists a dense and uncountable set of pairs $(\mathbf{x},\mathbf{y})\in\mathbb{R}^m\times \mathbb{R}^m \cong M_{m,2}$ such that the system of inequalities

$$\|q\mathbf{x} + \mathbf{y} - \textbf{p}\| \leqslant f(t) \quad \mathrm{and}\quad |q|\leqslant t$$

has a nonzero integer solution $(\textbf{p},q)$ for all large enough t. In fact, this statement follows from a old result by Khintchine [Reference KhintchineKhi47] which is not very well known. See also [Reference Moshchevitin and NeckrasovMN25, 3.3] for a discussion.

8.3 Rational approximations to linear subspaces

In this subsection we fix $d\in\mathbb{N}_{\geqslant2}$ and $a = 1,\dots, d-1$ , and consider a certain Diophantine system on $X = \operatorname{Gr}_{d,a}$ , the Grassmannian of a-dimensional subspaces of $\mathbb{R}^d$ . Following the set-up of [Reference SchmidtSch67], one can study the problems of approximation of an a-dimensional subspace A of $\mathbb{R}^d$ by b-dimensional rational subspaces B of $\mathbb{R}^d$ , where $1\leqslant b < d$ , in terms of the so-called first angle between the subspaces. The latter (or rather, formally speaking, the sine of the angle) is defined as follows:

$$\measuredangle_1 (A,B) {\, \stackrel{\mathrm{def}}{=}\, } \min_{\mathbf{x} \in A\smallsetminus\{0\},\, \mathbf{y} \in B\smallsetminus\{0\}}\frac{\|\mathbf{x}\wedge \mathbf{y}\|}{\|\mathbf{x}\|\cdot \|\mathbf{y}\|},$$

where $\|\cdot\|$ is the Euclidean norm on $\mathbb{R}^d$ and on $\bigwedge^2(\mathbb{R}^d)$ .

Note that $\measuredangle_1 (A,B) = 0$ if and only if $\dim(A\cap B) >0$ . Using the terminology introduced in [Reference NeckrasovNec22], let us say that $A\in \operatorname{Gr}_{d,a}$ is completely irrational if for any $(d-a)$ -dimensional rational subspace R we have $A\cap R = \{{0}\}$ . We also define the height H(B) of a rational subspace $B\subset \mathbb{R}^d$ of dimension b in the natural way as the covolume of the b-dimensional lattice $ B\cap \mathbb{Z}^d$ . Using the methods of this paper, namely Theorem 1.5, it is possible to prove the following theorem.

Theorem 8.2. Let $1\leqslant a,b < d$ with $\max(a,b) > 1$ . Then for any non-increasing $f: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ the set of completely irrational $A\in \operatorname{Gr}_{d,a}$ such that the system of inequalities

$$H(B) \leqslant t,\,\,\, \measuredangle_1 (A,B) \leqslant f (t)$$

has a solution in b-dimensional rational subspaces B for all large enough t is uncountable and dense.

The case $d=4$ and $a=b=2$ is a recent result of Chebotarenko [Reference ChebotarenkoChe24]. The above theorem, as well as several generalizations, will be proved in a forthcoming work.