2 Dirichlet’s theorem over number fields

In this section, we first state Dirichlet’s theorem over number fields, which is a slight modification of [Reference Burger8, Lemma 5.1]. Before stating Dirichlet’s theorem, we recall some notation and terminology from [Reference Burger8].

We take $K$ to be an algebraic number field of degree $d$ over $\mathbb Q.$ The collection of all nontrivial places of $K$ is denoted by $P_K$ . For any $v \in P_K$ , $K_v$ denotes the completion of $K$ with respect to the place $v$ . We know that for any $v \in P_K$ , $K_v$ is isomorphic to a finite extension of either $\mathbb R$ or $\mathbb Q_p$ for some prime $p$ . For any given $v \in P_K$ , if $v $ is an Archimedean place, we say that $v$ lies over infinity, denoted by $v|\infty $ . If $v$ is a non-Archimedean place, then $v$ is the extension of some $p$ -adic valuation. In this case, we say $v$ lies over the finite prime $p$ , denoted by $v|p$ :

• • $d_v=[K_v : \mathbb Q_v]$ denotes the local degree of $K_v$ for each place $v \in P_K$ ;

• • $\|\cdot \|_v$ denotes the absolute value on $K_v$ for each place $v \in P_K$ .

For each place $v\in P_K$ , we normalize the absolute value $\|\cdot \|_v$ as follows:

1. (1) if $v|p$ , then $\|p\|_v=p^{-1}$ ;

2. (2) if $v|\infty $ , then for $x\in K_v,\, \|x\|_v=|x|$ , where $|\cdot |$ is the Euclidean absolute value on $\mathbb R$ or $\mathbb C$ .

So, $\|\cdot \|_v$ extends the usual $p$ -adic absolute value if $v|p$ and the Euclidean absolute value if $v|\infty $ . Our second normalized absolute value $|\cdot |_v$ on $K_v$ is defined by

$$ \begin{align*}|x|_v=\|x\|_{v}^{d_v}.\end{align*} $$

Due to this normalization, we have the product formula:

(2-1) $$ \begin{align} \prod_{v \in P_K} |x|_v =1 \end{align} $$

for all $x \in K,\, x\neq 0.$

• • For $\vec {x}=(x_1,x_2,\ldots ,x_n) \in K_{v}^{n}$ , we extend our absolute values as follows:

  $$ \begin{align*}|\vec{x}|_v&=\max_{1\leq i \leq n} \{|x_i|_v\},\\\|\vec{x}\|_v&=\max_{1\leq i \leq n} \{\|x_i\|_v\}.\end{align*} $$

• • Let $v$ be a finite place of $K.$ We denote the maximal compact (open) subring of $K_v$ by $\mathcal O_v,$ that is,

  $$ \begin{align*}\mathcal O_v = \{x \in K_v : |x|_v \leq 1 \}.\end{align*} $$

• • A subset $R_v$ in $K_v^n$ is a $K_v$ -lattice if it is a compact open $\mathcal O_v$ -module in $K_v^n.$ It is easy to observe that $\mathcal O_v^n$ is a $K_v$ -lattice in $K_v^n.$

• • Let $S$ be a finite collection of places of $K$ containing all Archimedean places. We define the ring of $S$ -integers as

  $$ \begin{align*}\mathcal O_S =\{x \in K : x \in \mathcal O_v \ \text{for} \ \text{all} \ v \notin S\}.\end{align*} $$

• • The adele ring of $K$ is denoted by $K_{\mathbb A}$ . Here, $K$ is diagonally embedded inside $K_{\mathbb A}$ , so that we may view $K \subseteq K_{\mathbb A}$ by the natural diagonal map $\rho : K \rightarrow K_{\mathbb A}$ defined by

  $$ \begin{align*}\rho(\alpha)=(\alpha,\alpha,\alpha,\ldots )\end{align*} $$
  for $\alpha \in K.$

• • We say that a nonempty subset $R_v \subseteq K_v^n$ is a regular set if it has one of the following forms:

  1. (1) if $v|\infty $ , then $R_v$ is a bounded, convex, closed, symmetric subset of $K_v^n$ with nonzero volume;

  2. (2) if $v \nmid \infty $ , then $R_v$ is a $K_v$ -lattice in $K_v^n$ .

• • We say that a subset $\mathcal R $ of $K_{\mathbb A}^n$ is admissible if $\mathcal {R}= \prod _{v \in P_K} R_v,$ where $R_v$ is a regular set in $K_v^n$ for each $v \in P_K$ and $R_v=\mathcal {O}_v^n$ for almost all places $v.$

• • We define the field constant for the number field $K$ as

  $$ \begin{align*} \text{const}_K=\bigg(\frac{2}{\pi}\bigg)^s|D_K|^{1/2}, \end{align*} $$
  where $D_K$ is the discriminant of $K$ and $s$ is the number of complex places of $K.$

We are now ready to state and prove a version of Dirichlet’s theorem over number fields. The following theorem is a slight modification of [Reference Burger8, Lemma 5.1], in the sense that we are considering a multiplicative form of approximation. Although the proof follows from [Reference Burger8, Lemma 5.1], we include it here for completeness.

Theorem 2.1. Let $A_v$ be an $m \times n$ matrix over $K_v$ for each $v \in S.$ Also, let

$$ \begin{align*}\varepsilon_{v}^{(1)},\ldots,\varepsilon_{v}^{(m)},\delta_{v}^{(1)},\ldots,\delta_{v}^{(n)} \in K_v \setminus \{0\}\end{align*} $$

be given for each $v \in S$ so that $| \varepsilon _{v}^{(i)} |_v < 1, \text { for all } i=1, \ldots ,m$ and ${|\delta _{v}^{(j)}|_v \geq 1,}\, \text { for all } j=1,\ldots ,n$ for each $v \in S$ , and

$$ \begin{align*} \prod_{v \in S}| \varepsilon_{v}^{(1)} |_v \cdots | \varepsilon_{v}^{(m)} |_v |\delta_{v}^{(1)}|_v \cdots |\delta_{v}^{(n)}|_v=(\mathrm{const}_K)^{m+n}. \end{align*} $$

Then, there exists $\vec {x}=(x_1,\ldots ,x_n) \in \mathcal O_S^n \setminus \{\vec {0}\}$ and $\vec {y}=(y_1,\ldots ,y_m) \in \mathcal O_S^m$ satisfying

$$ \begin{align*} \left \{ \begin{array}{@{}rcl} \|A_{v}^{(i)} \vec{x} - y_i \|_v \leq \|\varepsilon_{v}^{(i)}\|_v &\text{for} \ i=1,2,\ldots,m, \\ \|x_j\|_v \leq \|\delta_{v}^{(j)}\|_v &\text{for} \ j=1,2,\ldots,n , \end{array} \right. \end{align*} $$

for all $v \in S,$ where $A_{v}^{(i)}$ is the $i$ th row of $A_v$ for $i=1,2,\ldots , m.$

Proof. The proof of this theorem goes along the lines of the original proof by Burger with slight modifications. For each place $v \in P_K,$ we define an $(m+n)\times (m+n)$ matrix $B_v$ over $K_v$ by

$$ \begin{align*} B_v=\begin{pmatrix} \begin{matrix} {\delta_v}^{-1}I_n \end{matrix} & {\mathbf{\large 0}} \\ \varepsilon_v^{-1}A_v & \begin{matrix} {\varepsilon_v}^{-1}I_m \end{matrix} \end{pmatrix} \end{align*} $$

for all $v \in S,$ where $\varepsilon _v^{-1}A_v$ is the $m \times n$ matrix over $K_v,$ whose $i$ th row is ${\varepsilon _v^{(i)}}^{-1}A_v^{(i)}$ for ${i=1, \ldots ,m;}\, {\delta _v}^{-1}I_n=\text {diag}({\delta _v^{(1)}}^{-1}, {\delta _v^{(2)}}^{-1}, \ldots , {\delta _v^{(n)}}^{-1})$ and we also have $ \varepsilon _v^{-1}I_m=\text {diag}({\varepsilon _v^{(1)}}^{-1}, {\varepsilon _v^{(2)}}^{-1},\, \ldots , {\varepsilon _v^{(m)}}^{-1}).$ Also, define $B_v=I_{m+n}$ for all $v \notin S.$

Define $R_v \subseteq K_{v}^{m+n}$ by

$$ \begin{align*}R_v=\{\vec{z} \in K_{v}^{m+n} : \|B_v \vec{z}\|_v \leq 1 \}\end{align*} $$

and let $\mathcal R=\prod _{v} R_v.$ It is easy to see that $\mathcal R \subseteq K_{\mathbb A}^{m+n}$ and $\mathcal R $ is admissible. Observe that

$$ \begin{align*} \displaystyle V(\mathcal R)& =2^{d(m+n)}\bigg(\frac{\pi}{2}\bigg)^{s(m+n)}|D_K|^{-(m+n)/2} \bigg(\!\prod_{v}|\text{det}\ B_v|_v\bigg)^{-1} \\ & \displaystyle =2^{d(m+n)} \bigg(\bigg(\frac{\pi}{2}\bigg)^{s}|D_K|^{-1/2}\bigg)^{m+n} \bigg(\!\prod_{v} | \varepsilon_{v}^{(1)} |_v \cdots | \varepsilon_{v}^{(m)} |_v |\delta_{v}^{(1)}|_v \cdots |\delta_{v}^{(n)}|_v\bigg) \\ & \displaystyle =2^{d(m+n)}. \end{align*} $$

Let $\lambda _1, \lambda _2, \ldots , \lambda _{(m+n)}$ be the successive minima of $\mathcal R \subseteq K_{\mathbb A}^{m+n}$ with respect to $K^{m+n}.$ Then, by the adelic successive minima theorem [Reference Bombieri and Vaaler6, Theorem 3], we know that

$$ \begin{align*}(\lambda_1 \lambda_2 \cdots \lambda_{m+n})^d V(\mathcal R) \leq 2^{d(m+n)}.\end{align*} $$

Thus, we have $\lambda _1 \leq 1,$ since $\lambda _n$ are increasing and $V(\mathcal R )=2^{d(m+n)}.$ Hence, there exists a point $\begin {pmatrix} \vec {x} \\ - \vec {y} \end {pmatrix}\, \in K^{m+n}\setminus \{0\} \cap \mathcal R.$ By our definition of $R_v$ for $v \notin S,$

$$ \begin{align*}\begin{pmatrix} \vec{x} \\ - \vec{y} \end{pmatrix} \in \mathcal{O}_{S}^{m+n}.\end{align*} $$

Also, for $v \in S ,$

$$ \begin{align*}\|A_{v}^{(i)} \vec{x} - y_i \|_v \leq \|\varepsilon_{v}^{(i)}\|_v\end{align*} $$

and

$$ \begin{align*}\|x_j\|_v \leq \|\delta_{v}^{(j)}\|_v\end{align*} $$

for $i=1,2, \ldots ,m$ and $j=1,2,\ldots ,n$ . In addition, it is easy to see that $\vec {x} \neq \vec {0}$ due to the product formula (2-1).

Example 2.3. Let $m=n=1$ , $K=\mathbb {Q}$ and $S=\{\infty \}$ . In this case, Theorem 2.1 asserts that for any $x \in \mathbb {R}$ and any $Q \ge 1$ , there exist integers $p,q \in \mathbb {Z}$ with $1 \le q \le Q$ such that

$$ \begin{align*} |qx + p| < \frac{\mathrm{const}_{\mathbb{Q}}}{Q}. \end{align*} $$

Here, $\mathrm {const}_{\mathbb {Q}} = 1$ and this value is optimal in the following sense.

There exists $x \in \mathbb {R}$ such that, for every $c < 1$ , the system

$$ \begin{align*} |qx + p| < \frac{c}{Q}, \quad 1 \le q \le Q, \end{align*} $$

has no solution in integers $p,q$ for infinitely many $Q$ .

Example 2.4. Let $m=n=1, K=\mathbb {Q}(i)$ , where $i$ is a root of $x^2+1=0$ , and take $S=\{\infty \}$ . Then, Theorem 2.1 gives: for any $\alpha \in \mathbb {C}$ and any $Q \ge 1$ , there exist $p,q \in \mathbb {Z}[i]$ with $1 \le |q| \le Q$ such that

$$ \begin{align*} |q\alpha + p| < \frac{\mathrm{const}_{\mathbb{Q}(i)}}{Q}. \end{align*} $$

In this case,

$$ \begin{align*} \mathrm{const}_{\mathbb{Q}(i)} = \frac{4}{\pi}. \end{align*} $$

However, unlike the real case, this constant is not optimal. In fact, one can show that the optimal constant is strictly smaller, namely [Reference Minkowski37, Satz LXII, Section 8, Ch. 6]

$$ \begin{align*} \frac{\sqrt{2}}{3-\sqrt{3}}. \end{align*} $$

More precisely, there exists $\alpha \in \mathbb {C}$ such that for every $c <(\!\!{\sqrt {2}}/({3-\sqrt {3}}))$ , the system

$$ \begin{align*} |q\alpha + p| < \frac{c}{Q}, \quad 1 \le |q| \le Q, \end{align*} $$

has no solution in $p,q \in \mathbb {Z}[i]$ for infinitely many $Q$ .

3 Dirichlet improvability and the main theorem

Before introducing the notion of ‘Dirichlet improvability’, we first recall some notation from [Reference Kleinbock and Tomanov27].

• • Throughout this article, $K$ represents a number field of degree $d$ .

• • We denote a finite set of places of $K$ containing all the Archimedean places as $S$ . Precisely, let $S=\{v_1,\ldots ,v_{\ell }\},$ where $v_i \in P_K$ for $i=1,\ldots ,\ell ,$ such that $S$ contains all Archimedean places.

• • We denote the normalized absolute value on $K_v$ by $|\cdot |_v,$ which is the same as the earlier defined normalization $|\cdot |_v$ on $K_v.$

• • $K_S$ is the direct product of all the completions $K_v,\, v \in S,$ that is,

  $$ \begin{align*}K_S=\prod_{v \in S}K_v.\end{align*} $$
  $K$ is embedded inside $K_S$ via the diagonal embedding. The set of all Archimedean places of $K$ is denoted by $S_a \subseteq S$ and we denote $S \setminus S_a $ by $S_f.$ Let $S_c$ and $S_r$ respectively denote the set of all complex places and the set of all real places of $K.$

• • The elements of $K_S$ are denoted as $x=(x^{(v)})_{v \in S},$ or simply $x=(x^{(v)})_v,$ where $x^{(v)} \in K_v.$ For $x \in K_S$ , we define the content $c(x)$ of $x$ as follows:

  $$ \begin{align*}c(x)=\prod_{v \in S}|x^{(v)}|_v .\end{align*} $$

• • As usual, the ring of integers of $K$ is denoted by $\mathcal O_K.$

• • The ring of $S$ -integers of $K$ , that is, $\mathcal O_S$ , is also diagonally embedded inside $K_S.$ We use this identification for convenience.

• • We denote the elements of $K_{S}^{m}$ by bold characters as $\mathbf {x}=(\mathbf {x}^{(v)})_v \in K_{S}^{m},$ where $\mathbf {x}^{(v)}=(x_{1}^{(v)},\ldots ,x_{m}^{(v)}) \in K_{v}^{m}.$

• • We denote by $\|\cdot \|_{v,2}$ the square of the usual Hermitian norm on $K_v^m$ if $v$ is complex, the usual Euclidean norm on $K_v^m$ if $v$ is real and the sup norm defined by

  $$ \begin{align*}\|(x_1^{(v)}, \ldots,x_m^{(v)})\|_{v,2} = \max_{i}|x_i^{(v)}|_v\end{align*} $$
  if $v$ is non-Archimedean.

  Note that if $v$ is non-Archimedean and $\vec {x}=(x_1,\ldots ,x_m) \in K_v^m,$ then $\|\vec {x}\|_{v,2}=|\vec {x}|_v.$

• • We endow both $K_S$ and $K_S^m$ with the product metric on them, respectively. More precisely, for any $x=(x^{(v)})_v \in K_S$ and $\mathbf {x}=( \mathbf {x}^{(v)})_v \in K_S^m$ , the norms are given by

  $$ \begin{align*}\|x\|:=\sup_{v\in S} |x^{(v)}|_v\end{align*} $$
  and
  $$ \begin{align*}\| \mathbf{x} \|:= \sup_{v\in S} \|\mathbf{x}^{(v)}\|_{v,2},\end{align*} $$
  respectively.

• • For $\mathbf {x}=(\mathbf {x}^{(v)})_v \in K_{S}^{m}$ , we define the content $c(\mathbf {x})$ of $\mathbf {x}$ by

  $$ \begin{align*}c(\mathbf{x})=\prod_{v\in S} \|\mathbf{x}^{(v)}\|_{v,2}.\end{align*} $$

• • For the sake of convenience, we keep the notation $\|\cdot \|_{v,2}$ and $c(\cdot )$ to denote the norms and the content on the exterior powers $\bigwedge ^r (K_v^m)$ and $\bigwedge ^r (K_S^m),$ respectively.

• • Given a field $F$ , the set of all $m \times n$ matrices over $F$ is denoted by $M_{m\times n}(F).$

• • We define $M_{m\times n}(K_S)$ as

  $$ \begin{align*} M_{m\times n}(K_S):=\prod_{v \in S} M_{m\times n}(K_v).\end{align*} $$
  Elements of $M_{m\times n}(K_S)$ are denoted by $g=(g^{(v)})_v,$ where $g^{(v)} \in M_{m\times n}(K_v).$ Similarly,
  $$ \begin{align*}\operatorname{\mathrm{GL}}(m,K_S):=\prod_{v \in S}\operatorname{\mathrm{GL}}(m,K_v)\end{align*} $$
  and
  $$ \begin{align*}\operatorname{\mathrm{GL}}^1(m,K_S):=\bigg\{\big(g^{(v)}\big)_{v \in S} \in \operatorname{\mathrm{GL}}(m,K_S) : \prod_{v\in S} \big|\text{det}\big(g^{(v)}\big)\big|_v =1\bigg\}.\end{align*} $$

• •

  $$ \begin{align*}\notag\\[-36pt] \begin{array}{ll} \mathfrak{a}^{+}:=&\!\!\! \{\mathbf{t}:=(t_{v_1}^{(1)},t_{v_1}^{(2)},\ldots,t_{v_1}^{(m+n)},\ldots,t_{v_{\ell}}^{(1)},t_{v_{\ell}}^{(2)},\ldots,t_{v_{\ell}}^{(m+n)} ) \in \mathbb{R}_{+}^{m+n} \times \cdots \times \mathbb{R}_{+}^{m+n} :\\& t_{v_r}^{(i)} < 1 , \,\, t_{v_r}^{(j)} \geq 1\quad \text{for} \,i=1,\ldots,m;\,\, j=m+1,\ldots,m+n;\,\, r=1,\ldots,\ell \\ & \text{and} \,\, \prod_{v\in S} t_{v}^{(1)} \cdots t_{v}^{(m+n)}=(\text{const}_K)^{m+n} \}, \end{array} \end{align*} $$
  where $\mathbb {R}^+$ is the set of positive real numbers.

• • Let $\mathcal {I}$ be a subset of $\mathfrak {a}^+$ defined by

  $$ \begin{align*} \begin{array}{rcl} \mathcal{I}:= \{\mathbf{t} \in \mathfrak{a}^+ : t_{v_r}^{(1)}=\cdots=t_{v_r}^{(m)}, t_{v_r}^{(m+1)}=\cdots=t_{v_r}^{(m+n)} \,\, \text{for} \,\, r=1,\ldots,\ell \}. \end{array} \end{align*} $$

• • Let $S$ contain precisely all Archimedian places of $K$ . Given $\mathbf {t} \in \mathcal {I},$ let ${t_{v_r}:=t_{v_r}^{(1)}=\cdots =t_{v_r}^{(m)}}$ and $t^{\prime }_{v_r}:=t_{v_r}^{(m+1)}=\cdots =t_{v_r}^{(m+n)}$ for $r=1,\ldots ,\ell .$ Keeping this in mind, let $\mathcal {I}_0$ be the subset of $\mathcal {I}$ defined as follows:

  $$ \begin{align*} \begin{array}{rcl} \mathcal{I}_0 := \{ \mathbf{t} \in \mathcal{I}:t_{v_r}=t_{v_s} \,\, \text{and} \,\, t^{\prime}_{v_r}=t^{\prime}_{v_s} \,\, \text{for}\,\, r,s=1,\ldots,\ell \}. \end{array} \end{align*} $$
  Here $\mathcal {I}_0$ is called the central ray of $\mathfrak {a}^+.$

• • For $ \mathbf {t} \in \mathfrak {a}^{+}, $ define $\|\mathbf {t}\|_{\infty }= \displaystyle \max _{\scriptsize {\begin {array}{rcl} i=1,\ldots ,m+n \\ r=1,\ldots ,\ell \end {array}}} t_{v_r}^{(i)}.$

• • For $i=1,\ldots ,\ell $ , let $p_i : \mathfrak {a}^{+} \rightarrow \mathbb {R}_{+}^{m+n}$ be the projection of $\mathfrak {a}^{+}$ on the $i$ th factor $\mathbb {R}_{+}^{m+n}$ defined by

  $$ \begin{align*}p_i(\mathbf{t})=(t_{v_i}^{(1)},t_{v_i}^{(2)},\ldots,t_{v_i}^{(m+n)}) \quad \text{for all } \,\, \mathbf{t} \in \mathfrak{a}^{+} .\end{align*} $$

• • We say that a subset $\mathcal {T}$ of $\mathfrak {a}^{+}$ is $S$ -unbounded if $p_i|_{\mathcal T}$ is unbounded for each $i=1,\ldots ,\ell .$

We define Dirichlet improvability in this context. Let $\mathfrak {a}^{+}$ be as above and $\mathcal T $ be an $S$ -unbounded subset of $\mathfrak {a}^{+}$ .

Definition 3.1. Given $A=(A_v)_{v}\in M_{m\times n}(K_S),$ where $A_v$ is an $m \times n$ matrix over $K_v$ for each $v\in S$ , and $0 < \varepsilon < 1$ , we say that Dirichlet’s theorem can be $\varepsilon $ -improved for $A$ along $ \mathcal {T} $ or use the notation $A \in DI_{\varepsilon }(m,n,K_S,\mathcal T)$ if there is $t_0>0$ such that given any $\varepsilon _{v}^{(1)}, \varepsilon _{v}^{(2)}, \ldots , \varepsilon _{v}^{(m)},\delta _{v}^{(1)}, \delta _{v}^{(2)}, \ldots , \delta _{v}^{(n)} \in K_v \setminus \{0\}$ for each $v \in S$ with

(3-1) $$ \begin{align} \mathbf{t}:= (& |\varepsilon_{v_1}^{(1)}|_{v_1},\ldots,|\varepsilon_{v_1}^{(m)}|_{v_1},|\delta_{v_1}^{(1)}|_{v_1} ,\ldots,|\delta_{v_1}^{(n)}|_{v_1} ,\ldots,|\varepsilon_{v_{\ell}}^{(1)}|_{v_{\ell}},\ldots,|\varepsilon_{v_{\ell}}^{(m)}|_{v_{\ell}}, \nonumber \\ & |\delta_{v_{\ell}}^{(1)}|_{v_{\ell}} ,\ldots,|\delta_{v_{\ell}}^{(n)}|_{v_{\ell}} ) \in \mathcal{T} \end{align} $$

and $\|p_i|_{\mathcal {T}}(\mathbf {t})\|_{\infty }> t_0$ for each $i=1,\ldots ,\ell ,$ there exists $\vec {x}=(x_1,\ldots ,x_n) \in \mathcal O_S^n \setminus \{\vec {0}\}$ and ${\vec {y}=(y_1,\ldots ,y_m) \in \mathcal O_S^m}$ satisfying

(3-2) $$ \begin{align}\left \{ \begin{array}{@{}rcl} \|A_{v}^{(i)} \vec{x} - y_i \|_v \leq \varepsilon \|\varepsilon_{v}^{(i)}\|_v &\text{for} \,i=1,2,\ldots,m ,\\ \|x_j\|_v \leq \varepsilon \|\delta_{v}^{(j)}\|_v &\text{for} \,j=1,2,\ldots,n\,, \end{array} \right. \end{align} $$

for all $v \in S,$ where $A_{v}^{(i)}$ is the $i$ th row of $A_v$ for $i=1,2,\ldots , m$ . In particular, a vector $ \mathbf {x}=(\mathbf {x}^{(v)})_v \in K_{S}^{n},$ where $\mathbf {x}^{(v)}=(x_1^{(v)},\ldots ,x_n^{(v)}) \in K_v^n$ is said to be Dirichlet $\varepsilon $ -improvable along $\mathcal {T}$ if the corresponding matrix $A=([x_1^{(v)} \cdots x_n^{(v)}])_{v \in S}\in DI_{\varepsilon }(1,n,K_S,\mathcal {T})$ . When $\mathcal {T}=\mathcal {I} $ , then simply we say $A$ is Dirichlet $\varepsilon $ -improvable.

Singular vectors can be defined in the $S$ -adic context in a similar way to that of the Euclidean context.

When $K$ is totally real and $S=$ the collection of all Archimedian places of $K$ , the notion of singular vectors has been introduced in [Reference Datta and Radhika13].

Definition 3.3. [Reference Datta and Radhika13, Definition 2.2] Let $K$ be a totally real number field and $S$ the collection of all Archimedian places of $K$ . We say $\mathbf {x}=(x_1,\ldots ,x_n) \in K_S^n$ is singular if for every $c>0,$ for all sufficiently large $Q>0$ , there exists $\mathbf {0} \neq \mathbf {q} \in \mathcal O_K^n,\, q_0 \in \mathcal O_K$ satisfying the following system:

(3-3) $$ \begin{align}\left \{ \begin{array}{@{}l} \| \mathbf{q} \cdot \mathbf{x} + q_0 \| < \dfrac{c}{Q^n}, \\ \qquad\quad\, \|\mathbf{q}\| \leq Q \,. \end{array} \right. \end{align} $$

The set of all singular vectors in $K_S^n $ , in this sense, is denoted by $Sing_S^n$ .

Indeed, when $S$ is the set of all Archimedian places of $K$ , Definition 3.3 is equivalent to our definition. The following proposition establishes that.

Proof. By a standard algebraic argument using fractional ideals and so forth, one can easily see that, in this case, $\mathcal O_S=\mathcal O_K$ . Let $\mathbf {x} \in Sing_S^n$ . We show that $\mathbf {x} \in Sing(K_S,n,\mathcal {I}_0).$ For that, let us take $0< \varepsilon < 1.$ Suppose that $\varepsilon _v,\, \delta _v\, \in K_v \setminus \{0\}$ , for each $v \in S$ with $|\varepsilon _v|_v=|\varepsilon _{v'}|_{v'}=\varepsilon _S,\, |\delta _v|_v=|\delta _{v'}|_{v'}=\delta _S$ for all $v,v'\in S$ and $ \varepsilon _S <1$ , $\delta _S \gg 1$ with

$$ \begin{align*} \varepsilon_S^{\ell} \delta_S^{n\ell}=(\text{const}_K)^{n+1}.\end{align*} $$

We choose $c>0$ and $Q$ sufficiently large so that

(3-4) $$ \begin{align} \left \{ \begin{array}{@{}rcl} \dfrac{c}{Q^n} < \varepsilon \varepsilon_S, \\[6pt] Q \leq \varepsilon \delta_S \,, \end{array} \right. \end{align} $$

holds. Since $\mathbf {x}$ is singular, for this chosen $c $ and sufficiently large $Q$ , there exists $\mathbf {0} \neq \mathbf {q} \in \mathcal O_K^n,\, q_0 \in \mathcal O_K $ such that (3-3) holds. Observe that if $\mathbf {x}=(x_1,\ldots ,x_n) \in K_S^n$ and $x_i=(x_i^{(v)}) \in K_S$ for $i=1,\ldots ,n,$ then (3-3) can be rewritten as

(3-5) $$ \begin{align} | q_1 x_1^{(v)}+q_2 x_2^{(v)}+ \cdots +q_n x_n^{(v)} + q_0|_v \leq \frac{c}{Q^n} \quad\text{and}\quad \displaystyle \max_{1\leq j \leq n} |q_j|_v \leq Q \, \end{align} $$

for all $v \in S$ .

Combining (3-4) and (3-5), we get that $\mathbf {x} \in K_S^n$ is Dirichlet $\varepsilon $ -improvable along $\mathcal {I}_0$ . Therefore,

$$ \begin{align*}Sing_S^n \subseteq Sing(K_S,n,\mathcal{I}_0).\end{align*} $$

The other containment $Sing(K_S,n,\mathcal {I}_0) \subseteq Sing_S^n$ is obvious.

Unless specified otherwise, we henceforth let $\mathbf {t}$ stand for (3-1). Before going to our main result, let us recall some terminology from [Reference Kleinbock, Lindenstrauss and Weiss24, Reference Kleinbock and Tomanov27, Reference Kleinbock and Tomanov28].

Definition 3.6. Let $X$ be a metric space. We say that $X$ is $\textit {Besicovitch}$ if for any bounded subset $A$ of $X$ and for any family $\mathcal {B}$ of nonempty open balls in $X$ such that every $x \in A$ is the centre of some open ball of $\mathcal {B}$ , there exists a constant $N_X$ (called the Besicovitch constant of $X$ ) and there is a finite or countable subfamily $\{B_i\}$ of $\mathcal {B}$ such that

$$ \begin{align*}1_A\leq \displaystyle \sum_i 1_{B_i} \leq N_X\,,\end{align*} $$

that is, $A \subset \bigcup _i B_i$ , and the multiplicity of that subcovering is at most $N_X$ .

Throughout this article, $N_X$ always represents the Besicovitch constant of the metric space $X$ .

Let $X$ be a Besicovitch metric space and $(F,|\cdot |)$ a valued field. For a subset $B$ of $X$ and a given function $f : X \rightarrow F$ , we set

$$ \begin{align*}\|f\|_B := \displaystyle \sup_{x\in B} |f(x)|.\end{align*} $$

If $f: X \rightarrow F$ is a measurable function, $\mu $ is a Radon measure on $X$ and $B$ is a subset of $X$ with $\mu (B)>0$ , we define

$$ \begin{align*}\|f\|_{\mu,B} :=\|f\|_{B\cap \text{ supp }(\mu)} .\end{align*} $$

Definition 3.8. Given $C,\alpha>0$ , open $U \subset X $ and a Radon measure $\mu $ on $X$ , $f: X \rightarrow F$ is said to be $(C,\alpha )$ -good on $U$ with respect to $\mu $ if for every ball $B\subset U$ with centre in $\text {supp }(\mu )$ and any $\varepsilon>0$ ,

$$ \begin{align*}\mu(\{x\in B: |f(x)| < \varepsilon\})\leq C\bigg(\frac{\varepsilon}{||f||_{\mu,B}}\bigg)^{\alpha} \mu(B)\,.\end{align*} $$

The properties listed below follow immediately from Definition 3.8.

We recall an important lemma from [Reference Kleinbock and Tomanov27, Section 2.3], which is useful later in our discussion.

Given a Radon measure $\mu $ on $X$ , an open subset $U$ of $X$ with $\mu (U)>0$ and a map $\mathbf {f}=(f_1,f_2,\ldots ,f_m) : U \rightarrow F^m$ , we say that the pair $(\mathbf {f},\mu )$ is $(C,\alpha )$ -good on $U$ if any $F$ -linear combination of $ 1,f_1,\ldots ,f_m $ is $(C,\alpha )$ -good on $U$ with respect to $\mu .$

If $B=B(x,r),$ where $x\in X$ and $r>0$ , is a ball in $X$ and $c>0$ , we use the notation $cB$ to denote the ball $B(x,cr).$

Definition 3.12. Let $\mu $ be a Radon measure on $X$ and $D>0.$ Here, $\mu $ is said to be $D$ -Federer on an open subset $U$ of $X$ if for all balls $B$ centred at $\text { supp }(\mu )$ with $3B\subset U$ , one has

$$ \begin{align*}\frac{\mu(3B)}{\mu(B)}\leq D\,.\end{align*} $$

We finish this section by stating the main result of this paper, which says that for sufficiently small $\varepsilon>0$ , $\mu $ -almost every vector lying in the image of a good and nonplanar map is not Dirichlet $\varepsilon $ -improvable along $\mathcal {T}$ for a large class of measures $\mu $ in this $S$ -adic setup.

Theorem 3.13. Let $X= \prod _{v \in S}X_v,$ where $X_v=K_{v}^{l_v}$ with $l_v \in \mathbb N$ and $v\in S$ is a Besicovitch metric space, $\mu _v$ be a measure on $X_v\, (v \in S)$ so that $\mu =\prod _{ v \in S} \mu _v$ is $D$ -Federer measure on $X,$ and $U=\prod _{v \in S} U_v \subseteq X$ be open with $\mu (U)>0$ . Let $ \mathbf {f}=(\mathbf {f}^{(v)})_v:U \rightarrow K_S^n$ be a map, where $\mathbf {f}^{(v)}$ are continuous maps from $U_v \subseteq K_{v}^{l_v}$ to $K_v^n$ such that $(\mathbf {f}^{(v)},\mu _v)$ is nonplanar on $U_v$ for all $v \in S$ and for each $v\in S,\, (\mathbf {f}^{(v)},\mu _v)$ is $(C_v,\alpha _v)$ -good on $U_v.$ Then, there exists $\varepsilon _0=\varepsilon _0(n,C_{v_1}, \ldots , C_{v_\ell }, \alpha _{v_1}, \ldots , \alpha _{v_\ell }, D,N_X,K,S)>0$ such that for any $\varepsilon <\varepsilon _0$ ,

$$ \begin{align*} \mathbf{f}_{*}\mu(DI_{\varepsilon}(1,n,K_S,\mathcal{T}))=0 \,\,\text{ for any S-unbounded }\mathcal{T}\subseteq \mathfrak{a}^{+}\,. \end{align*} $$

Corollary 3.14. Let the hypothesis be the same as that of Theorem 3.13. Then,

$$ \begin{align*}\mathbf{f}_* \mu (Sing(K_S,n,\mathcal{T}))=0 \quad \text{for any S-unbounded }\mathcal{T}\subseteq \mathfrak{a}^{+}.\end{align*} $$

In particular, $\mathbf {f}_* \mu (Sing(K_S,n,\mathcal {I}_0))=0.$

In view of Lemma 3.4, one sees that Corollary 3.14 generalizes [Reference Datta and Radhika13, Theorem 2.3] to the case $X=\prod _{v \in S} K_{v}^{l_v}$ with $l_v \in \mathbb N$ for $v\in S$ .

The key idea to prove Theorem 3.13 is the so-called ‘quantitative nondivergence’ technique, a generalization of nondivergence of unipotent flows on homogeneous spaces. To apply this technique, first we need to translate our problem related to Dirichlet $\varepsilon $ -improvability into certain properties of flows on some homogeneous spaces, which is the main theme of our next section.

4 The correspondence

Define

$$ \begin{align*}\Omega_{S,m}:= \{g\mathcal O_S^m : g \in \operatorname{\mathrm{GL}}(m,K_S)\}\end{align*} $$

and

$$ \begin{align*}\Omega_{S,m}^1:= \big\{\Lambda \in \Omega_{S,m} : \text{cov}(\Lambda)=(\!\sqrt{D_K})^m \big\}.\end{align*} $$

Let $G:= \operatorname {\mathrm {GL}}^1(m,K_S)$ and $\Gamma :=\operatorname {\mathrm {GL}}(m,\mathcal O_S),$ and let $\pi : G \rightarrow G/\Gamma $ be the quotient map. Then, $G$ acts on $G/\Gamma $ by left translation via the rule $g\pi (h)=\pi (gh)$ for $g,h \in G.$ We define a map $\tau : M_{m \times n}(K_S) \longrightarrow \operatorname {\mathrm {GL}}^1(m+n,K_S)$ given by

$$ \begin{align*}\tau ((A_v)_{v })=(g_{A_v})_{v },\end{align*} $$

where

$$ \begin{align*}g_{A_v}:= \begin{bmatrix} I_m& A_v\\0& I_n \end{bmatrix}.\end{align*} $$

We put $\overline {\tau }:=\pi \circ \tau .$ The homogeneous space $\operatorname {\mathrm {GL}}^1(m,K_S)/\operatorname {\mathrm {GL}}(m,O_S)$ is naturally identified with $\Omega _{S,m}^1,$ so that we have $\overline {\tau }((A_v)_{v})=(g_{A_v})_{v } \mathcal O_S^{m+n}.$ Define

$$ \begin{align*}\Omega_{\varepsilon} :=\{ \Lambda \in \Omega_{S,m+n}:\delta(\Lambda) \geq \varepsilon\},\end{align*} $$

where $\delta : \Omega _{S,m+n} \longrightarrow \mathbb R_{+} $ is defined by

$$ \begin{align*}\delta(\Lambda) :=\min \{c(\mathbf{x}): \mathbf{x} \in \Lambda \setminus \{0\}\}\end{align*} $$

for all $\Lambda \in \Omega _{S,m+n}.$

Again, we let $\varepsilon _{v}^{(1)}, \varepsilon _{v}^{(2)}, \ldots , \varepsilon _{v}^{(m)},\delta _{v}^{(1)}, \delta _{v}^{(2)}, \ldots , \delta _{v}^{(n)} \in K_v \setminus \{0\}$ for each $v \in S$ and define $g_{\varepsilon _v,\delta _v}$ by

$$ \begin{align*}g_{\varepsilon_v,\delta_v}:=\text{diag}\big(\big(\varepsilon_{v}^{(1)}\big)^{-1},\ldots,\big(\varepsilon_{v}^{(m)}\big)^{-1},\big(\delta_{v}^{(1)}\big)^{-1},\ldots,\big(\delta_{v}^{(n)}\big)^{-1}\big).\end{align*} $$

Then, clearly $(g_{\varepsilon _v,\delta _v})_{v } \in \operatorname {\mathrm {GL}}(m+n,K_S).$

The following proposition, which may be regarded as an analogue of ‘Dani correspondence’ over number fields, helps us to convert our problem regarding Dirichlet improvability to a problem of homogeneous dynamics.

Proposition 4.1. Let $0<\varepsilon <1$ , and suppose that $ A=\left(A_v\right)_{v} \in DI_{\varepsilon}\left(m,n,K_S,\mathcal T \right)$ . For each $v \in S$ , choose $\varepsilon_{v}^{(1)}, \varepsilon_{v}^{(2)}, \dots , \varepsilon_{v}^{(m)},\delta_{v}^{(1)}, \delta_{v}^{(2)}, \dots, \delta_{v}^{(n)} \in K_v \setminus \{0\}$ with $\mathbf{t}\in \mathcal T$ given by (3-1) and $\|p_i|_{\mathcal{T}}(\mathbf{t})\|_{\infty} \gg 1$ . Then

$$ \begin{align*}\left(g_{\varepsilon_v,\delta_v}\right)_{v} \left(g_{A_v}\right)_{v} \mathcal O_S^{m+n} \notin \Omega_{\left(m+n\right)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon} . \end{align*} $$

Proof. Let $A=(A_v)_{v} \in DI_{\varepsilon }(m,n,K_S,\mathcal T).$ Then, given any $\varepsilon _{v}^{(1)}, \varepsilon _{v}^{(2)}, \ldots $ , $\varepsilon _{v}^{(m)},\delta _{v}^{(1)}, \delta _{v}^{(2)}, \ldots , \delta _{v}^{(n)}$ in $ K_v \setminus \{0\}$ for $v\in S$ , with $\mathbf {t} \in \mathcal T$ , where $\mathbf {t}$ is given by (3-1), and $\|p_i|_{\mathcal {T}}(\mathbf {t})\|_{\infty }>>1 $ for each $i=1,\ldots ,\ell $ , there exists $\vec {x}=(x_1,\ldots ,x_n) \in \mathcal O_S^n \setminus \{0\}$ and $\vec {y}=(y_1,\ldots ,y_m) \in \mathcal O_S^m$ such that (3-2) holds. Observe that

$$ \begin{align*} \begin{array}{rl} (g_{\varepsilon_v,\delta_v})_{v} (g_{A_v})_{v} \mathcal O_S^{m+n}&\!\!\!= \{ ( {\varepsilon_{v}^{(1)}}^{-1}(A_v^{(1)} \vec{b} - a_1), \ldots ,{\varepsilon_{v}^{(m)}}^{-1}(A_v^{(m)} \vec{b} - a_m), \\ & \quad {\delta_{v}^{(1)}}^{-1} b_1,\ldots, {\delta_{v}^{(n)}}^{-1} b_n)_{v } : \mathbf{z}=(\mathbf{z}^{(v)})_{v }\in \mathcal O_S^{m+n} \} \,,\end{array} \end{align*} $$

where $\mathbf {z}^{(v)}=(\vec {a},\vec {b}) \in K^{m+n}$ for each $v\in S,\, \vec {a}=(a_1,\ldots ,a_m) \in \mathcal {O}_S^m$ and $\vec {b}=(b_1,\ldots ,b_n) \in \mathcal {O}_S^n.$

Consider $\vec {x}\in \mathcal O_S^n \setminus \{0\}$ and $\vec {y} \in \mathcal O_S^m$ satisfying (3-2) as elements of $ K_S^n \setminus \{0\}$ and $K_S^m$ , respectively, and denote them as $\mathbf x=(\mathbf {x}^{(v)})_{v} \in \mathcal O_S^n \setminus \{0\}$ and $\mathbf {y}=(\mathbf {y}^{(v)})_{v } \in \mathcal O_S^m $ , respectively. Then, $(\mathbf {x},\mathbf {y}) \in \mathcal O_S^{m+n}$ and for this point, let $\mathbf {w}=(\mathbf {w}^{(v)})_{v}$ be the corresponding point on the lattice $(g_{\varepsilon _v,\delta _v})_{v} (g_{A_v})_{v} \mathcal O_S^{m+n}$ . Using (3-1) and (3-2),

$$ \begin{align*} \displaystyle c(\mathbf{w})= \prod_{v \in S } \|(\mathbf{w}^{(v)})\|_{v,2 } &\displaystyle = \bigg(\prod_{v \in S_r}\|\mathbf{w}^{(v)}\|_{v,2} \bigg)\, \bigg(\prod_{v \in S_c}\|\mathbf{w}^{(v)}\|_{v,2}\bigg)\, \bigg(\prod_{v \in S_f}\|\mathbf{w}^{(v)}\|_{v,2}\bigg) \\& \displaystyle < \bigg(\prod_{v \in S_r} \sqrt{(m+n){\varepsilon}^2} \bigg)\, \bigg(\prod_{v \in S_c} (m+n){\varepsilon}^2 \bigg) \bigg(\prod_{v \in S_f} \varepsilon\bigg) \\& \displaystyle = ((m+n)^{|S_r|/2} \varepsilon^{|S_r|}) ((m+n)^{|S_c|} \varepsilon^{ 2|S_c|}) (\varepsilon^{|S_f|}) \\ & \displaystyle = (m+n)^{|S_r|/2 + |S_c|} \varepsilon^{|S_r|+2|S_c|+|S_f|} \\& \displaystyle < (m+n)^{|S_r|/2 + |S_c|} \varepsilon, \end{align*} $$

since $d_v \geq 1,\, |S_r|+2|S_c|+|S_f| \geq |S| \geq 1$ and $0 < \varepsilon <1,$ where for any finite set $A,\, |A|$ denotes the number of elements of $A.$

Therefore, $ \delta (\Lambda ) < (m+n)^{|S_r|/2 + |S_c|} \varepsilon ,$ where $\Lambda =(g_{\varepsilon _v,\delta _v})_{v} (g_{A_v})_{v} \mathcal O_S^{m+n} $ and so we finally get our desired result.

Using the above proposition, we can say that given a Besicovitch metric space $X,$ a Federer measure $\mu $ on $X,$ an open subset $U$ of $X$ with $\mu (U)>0$ and a map $F: U \longrightarrow M_{m \times n}(K_S)$ to prove $F_{*} \mu (DI_{\varepsilon }(m,n,K_S,\mathcal T))=\mu (F^{-1}(DI_{\varepsilon }(m,n,K_S,\mathcal T)))=0$ , it is enough to show that

(4-1) $$ \begin{align} \mu(F^{-1}(\{(A_v)_{v} \in M_{m \times n}(K_S) : (g_{\varepsilon_v,\delta_v})_{v} (g_{A_v})_{v} \mathcal O_S^{m+n} \notin \Omega_{(m+n)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon}\}))=0 \end{align} $$

for all $\varepsilon _{v}^{(1)}, \varepsilon _{v}^{(2)}, \ldots , \varepsilon _{v}^{(m)},\delta _{v}^{(1)}, \delta _{v}^{(2)}, \ldots , \delta _{v}^{(n)} \in K_v \setminus \{0\}$ (for each $v\in S$ ) with $\mathbf {t} \in \mathcal T$ and $\|p_i|_{\mathcal {T}}(\mathbf {t})\|_{\infty }>>1 $ for each $i=1,\ldots ,\ell ,$ where $\mathbf {t}$ is given by (3-1).

We suppose that for any ball $B \subseteq U$ centred in $\text {supp}\,(\mu )$ , there exists a $c\in (0,1)$ such that

(4-2) $$ \begin{align}\begin{aligned} & \displaystyle \mu (B \cap F^{-1}(\{(A_v)_{v} \in M_{m \times n}(K_S) : (g_{\varepsilon_v,\delta_v})_{v} \overline{\tau}((A_v)_{v}) \notin \Omega_{(m+n)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon}\})) \\& \qquad \displaystyle = \mu(\{\mathbf{x} \in B : (g_{\varepsilon_v,\delta_v})_{v} \overline{\tau}(F(\mathbf{x})) \notin \Omega_{(m+n)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon}\}) \leq c \mu(B) \end{aligned}\end{align} $$

holds for all $\varepsilon _{v}^{(i)}$ and $\delta _{v}^{(j)}$ satisfying the aforementioned conditions.

If (4-2) holds true, then for any ball $B \subseteq U$ centred in $\text {supp}\,(\mu )$ ,

(4-3) $$ \begin{align} & \frac{\mu (B \cap F^{-1} (\{(A_v)_{v} \in M_{m \times n}(K_S) : (g_{\varepsilon_v,\delta_v})_{v} \overline{\tau}((A_v)_{v}) \notin \Omega_{ (m+n)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon}\}))}{\mu(B)} \nonumber \\ & \qquad \leq \frac{c\mu(B)}{\mu(B)}=c<1 \end{align} $$

for all $\varepsilon _{v}^{(1)}, \varepsilon _{v}^{(2)}, \ldots , \varepsilon _{v}^{(m)},\delta _{v}^{(1)}, \delta _{v}^{(2)}, \ldots , \delta _{v}^{(n)} \in K_v \setminus \{0\}$ (for each $v\in S$ ) with $\mathbf {t} \in \mathcal T$ and $\|p_i|_{\mathcal {T}}(\mathbf {t})\|_{\infty }>>1 $ for each $i=1,\ldots ,\ell .$

Since (4-3) holds for any ball $B \subseteq U,$ it follows that no point $\mathbf {x} \in U \cap \text {supp}(\mu )$ is a point of density of the sets

$$ \begin{align*} F^{-1}(\{(A_v)_{v} \in M_{m \times n}(K_S) : (g_{\varepsilon_v,\delta_v})_{v} \overline{\tau}((A_v)_{v}) \notin \Omega_{ (m+n)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon}\})\end{align*} $$

for all $\varepsilon _{v}^{(1)}, \varepsilon _{v}^{(2)}, \ldots , \varepsilon _{v}^{(m)},\delta _{v}^{(1)}, \delta _{v}^{(2)}, \ldots , \delta _{v}^{(n)} \in K_v \setminus \{0\}$ (for each $v\in S$ ) with $\mathbf {t} \in \mathcal T$ and $\|p_i|_{\mathcal {T}}(\mathbf {t})\|_{\infty }>>1 $ for each $i=1,\ldots ,\ell ,$ which gives us (4-1), because of the following version of the Lebesgue density theorem for our concerned metric space $X=\prod _{v\in S} K_v^{l_v},$ where $l_v \in \mathbb N,$ endowed with the product metric.

Lemma 4.2. Let $X=\prod _{v\in S} K_v^{l_v},$ where $l_v \in \mathbb N$ , be endowed with the product metric. If $\mu $ is a Radon measure on $X$ and $\Omega $ is any measurable subset of $X,$ then almost every point $\mathbf {x}\in \Omega \,\cap \text { supp }(\mu )$ is a point of density of $\Omega $ , that is,

$$ \begin{align*} \displaystyle \lim_{\scriptsize{\begin{array}{rcl}\mu (B)\rightarrow 0\\ \mathbf{x}\in B\end{array}}}\frac{\mu(B\cap \Omega)}{\mu(B)}=1\,.\end{align*} $$

The proof of Lemma 4.2 goes along the same lines as the Lebesgue density theorem for Euclidean spaces with some appropriate modifications. Hence, we leave it to the reader.

As the space of our interest $X$ is locally compact, Hausdorff and second countable, to prove Theorem 3.13, it is enough to prove the theorem locally, that is, after choosing a suitable $\varepsilon _0>0$ , it is enough to show that for all $\mathbf {w} \in U \cap \text { supp }(\mu ),$ there exists a ball $B' \subseteq U$ containing $\mathbf {w}$ such that

$$ \begin{align*} \mu(B' \cap \mathbf{f}^{-1}(DI_{\varepsilon}(1,n,K_S,\mathcal{T})))=\mu(\{\mathbf{x} \in B': \mathbf{f}(\mathbf{x}) \in DI_{\varepsilon}(1,n,K_S,\mathcal{T})\})=0 \end{align*} $$

for all $\varepsilon < \varepsilon _0.$ From the above discussion, it follows that Theorem 3.13 is an immediate consequence of the following proposition.

Proposition 4.3. Fix $n \in \mathbb N$ , and for each $v \in S$ , let $l_v \in \mathbb N$ . Set $ X_v:=K_v^{l_v}$ and $X:=\prod_{v\in S}X_v $ . For each $v \in S$ , suppose that $\mu_v$ is a measure on $X_v$ , and let $\mu:=\prod_{v \in S}\mu_v$ be a Radon measure on X. Let $\mathbf x=(x_v)_v \in \mathrm{supp}\,(\mu)$ and $r>0$ . For each $v\in S$ , define $B_v:=B(x_v,r)$ , $\tilde{B_v}:=B(x_v,3^{n+1}r)$ . Set

$$ \begin{align*} B:=\prod_{v\in S}B_v\quad \text{and} \quad\tilde B:=\prod_{v\in S}\tilde B_v=3^{n+1}B. \end{align*} $$

Suppose that $\mu$ is D-Federer on $\tilde B$ . For every $v\in S$ , let $\mathbf f^{(v)}=\bigl(f_1^{(v)},\ldots,f_n^{(v)}\bigr):\widetilde B_v \longrightarrow K_v^n$ be continuous, and define $\mathbf f:=(\mathbf f^{(v)})_v:\widetilde B\longrightarrow K_S^n$ . Suppose that, for every $v\in S$ , there exist $C_v,\alpha_v>0$ such that:

1. (i) any $K_v$ -linear combination of $1,f_1^{(v)},\ldots,f_n^{(v)}$ is $(C_v,\alpha_v)$ good on $\tilde{B}_v$ with respect to $\mu_v$ for each $v \in S$ ; and

2. (ii) the restrictions of $1,f_1^{(v)},\ldots,f_n^{(v)}$ to $B(x_v,r) \cap \mathrm{ supp }(\mu_v)$ are linearly independent over $K_v$ for each $ v\in S $ .

Then there exist constants $\alpha, \rho, \widetilde C>0$ with the following property. For each $v\in S$ , choose $\varepsilon_v^{(1)},\ldots,\varepsilon_v^{(m)},\delta_v^{(1)},\ldots,\delta_v^{(n)}\in K_v\setminus \{0\}$ . Let $\mathbf t \in\mathcal T $ be given by (3-1), and suppose that $\|p_i|_{\mathcal T}(\mathbf t)\|_{\infty} \gg 1$ for each $i=1,\ldots,\ell$ . Then for any $0<\varepsilon\le \rho/\sqrt{D_K}$ ,

(4-4) $$ \begin{align} \mu\big(\big\{\mathbf{x}\in B\,:\, \big(g_{\varepsilon_v,\delta_v}\big)_{v} \overline{\tau}\big(\mathbf{f}(\mathbf{x})\big) \notin \Omega_{(n+1)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon} \big\}\big)\leq \tilde C \varepsilon ^{\alpha} \mu(B), \end{align} $$

where the constants $\rho$ and $\widetilde C$ are given explicitly by (5-4) and (5-5), respectively.

Theorem 3.13 follows immediately from Proposition 4.3. We choose our $ 0 < \varepsilon _0 \leq ({\rho }/{\sqrt {D_K}})$ so that $\tilde {C} \varepsilon _{0}^{\alpha } < 1.$ In view of the expression for $\tilde {C}$ , (5-5), it is easy to see that $\varepsilon _0$ depends only on $(n, C_{v_1}, \ldots , C_{v_\ell }, \alpha _{v_1}, \ldots , \alpha _{v_\ell },D,N_X,K,S).$ Consider $\mathbf {x} \in U \cap \text {supp}(\mu )$ and choose a ball $B'$ containing $\mathbf {x}$ such that $B' \subseteq \tilde {B'} :=3^{n+1}B' \subseteq U.$ Next, choose any ball $B \subseteq B'$ with centre in $\text {supp}(\mu )$ and consider $\tilde {B}:=3^{n+1}B.$ Since by the hypothesis of Theorem 3.13, for each $v \in S,\, (\mathbf {f}^{(v)},\mu _v)$ is nonplanar and $(C_v,\alpha _v)$ -good on $U_v$ with respect to $\mu _v,$ Conditions (i) and (ii) of Proposition 4.3 hold immediately. We set $c=\tilde {C}\varepsilon _{0}^{\alpha } <1$ and observe that (4-2) holds whenever $0 < \varepsilon < \varepsilon _0,$ in view of Proposition 4.3. This completes the proof of Theorem 3.13, assuming Proposition 4.3.

We need to prove Proposition 4.3. We do it using the so-called ‘quantitative nondivergence’ technique and our next section is all about that.

5 Quantitative nondivergence and the proof of Proposition 4.3

We first recall the $S$ -arithmetic quantitative nondivergence theorem from [Reference Kleinbock and Tomanov27].

5.1 Quantitative nondivergence

We denote the set of all primitive submodules of $\mathcal O_S^m$ by ${\mathfrak {P}}(\mathcal O_S,m).$ We recall the following theorem from [Reference Kleinbock and Tomanov27, Section 6.3].

Theorem 5.1. Let $X$ be a Besicovitch metric space and $\mu $ be a $D$ -Federer measure on $X.$ For $m \in \mathbb N,$ let a ball $B \subseteq X$ and a continuous map $h : \tilde B \longrightarrow \operatorname {\mathrm {GL}}(m,K_S)$ be given, where $\tilde B := 3^m B.$ Suppose that for some $C, \alpha>0$ and $0 < \rho <1 $ , one has:

1. (C1) for every $\Delta \in {\mathfrak {P}}(\mathcal O_S,m),$ the function $\mathrm{cov}(h(\cdot )\,\Delta )$ is $(C,\alpha )$ -good on $\tilde B$ with respect to $\mu ;$

2. (C2) for every $\Delta \in {\mathfrak {P}}(\mathcal O_S,m),\, \|\mathrm{cov}(h(\cdot ))\,\Delta \|_{\mu ,B} \geq \rho .$

Then, for any positive $\varepsilon \leq \rho / \sqrt {D_K}$ ,

(5-1) $$ \begin{align} \displaystyle \mu(\{x\in B\,:\, \delta(h(x)\mathcal O_S^m)\, <\, \varepsilon \,\,\})\leq mC(N_X D^2)^m \bigg(\frac{\varepsilon \sqrt{D_K}}{\rho}\bigg)^{\alpha} \mu(B)\,. \end{align} $$

For more details and the proof of Theorem 5.1, see [Reference Kleinbock and Tomanov27].

5.2 The proof of Proposition 4.3

From the definition of $\Omega _{\varepsilon }$ , it follows that

$$ \begin{align*} (g_{\varepsilon_v,\delta_v})_{v} \overline{\tau}( \mathbf{f}(\mathbf{x})) \notin \Omega_{(n+1)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon} \Longleftrightarrow \delta ((g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \mathcal O_S^{n+1}) < (n+1)^{|S_r|/2 + |S_c|} \displaystyle \varepsilon \,. \end{align*} $$

We use Theorem 5.1 in the setting

$$ \begin{align*}\begin{array}{c} X=\prod_{v \in S} X_v=\prod_{v \in S}K_v^{l_v},m=n+1;\\ \mu, B, C_v, \alpha_v \,\,\text{and}\,\,D\,\,\text{as in Proposition 4.3};\\ h(\mathbf{x})= (g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \text{ for all } \mathbf{x}=(x_v)_v\in \tilde{B}\,,\,\end{array} \end{align*} $$

where

$$ \begin{align*}g_{\varepsilon_v,\delta_v}:=\text{diag}((\varepsilon_{v}^{(1)})^{-1},(\delta_{v}^{(1)})^{-1}, (\delta_{v}^{(2)})^{-1} ,\ldots,(\delta_{v}^{(n)})^{-1}) \in \operatorname{\mathrm{GL}}(n+1,K_v),\end{align*} $$

with $\varepsilon _{v}^{(i)}$ and $\delta _{v}^{(j)}$ satisfying the aforementioned conditions and

$$ \begin{align*}g_{\mathbf{f}^{(v)}(x_v)}:= \begin{bmatrix} 1 & \mathbf{f}^{(v)}(x_v) \\0 & I_n \end{bmatrix} \in \operatorname{\mathrm{GL}}(n+1,K_v).\end{align*} $$

To get our desired result, (4-4), we just need to verify the two conditions (C1) and (C2) of Theorem 5.1. Observe that to check Conditions (C1) and (C2), we first need explicit expressions for functions $\mathbf {x} \mapsto \text {cov} (h(\mathbf {x}) \Delta )$ in our setting.

We recall that given any $\Delta \in {\mathfrak {P}}(\mathcal O_S,n+1)$ of rank $j$ , there exists an element $\mathbf {w} \in \bigwedge ^j (\mathcal {O}_S^{n+1})$ such that

$$ \begin{align*}\text{cov}(\Delta)=a_{K} c(\mathbf{w}),\end{align*} $$

where $a_{K} \in \mathbb R$ is a constant depending on $K.$ Therefore, we have $\text {cov}(h(\textbf {x})\Delta )=a_{K}c(h(\mathbf {x})\mathbf {w}).$

Before proceeding further, let us fix some notation. We take $\mathbf {e}_0,\mathbf {e}_1, \ldots , \mathbf {e}_n \in K_S^{n+1}$ as the standard basis of $K_S^{n+1}$ over $K_S,$ where

$$ \begin{align*}\mathbf{e}_i=( \mathbf{e}_{i}^{(v)})_v, \,\, \,\, \mathbf{e}_{i}^{(v)} \in K_v^{n+1} \quad \text{for each} \,\, i=0,\ldots,n ;\end{align*} $$

and $\{ \mathbf {e}_{i}^{(v)}\}_{i=0}^{n}$ forms the standard basis of $K_v^{n+1}$ over $K_v$ for each $v\in S.$ Let $I=\{i_1,i_2,\ldots ,i_j\} \subset \{0,1, \ldots ,n \}$ with $i_1 < i_2 < \cdots < i_j $ and $\mathbf {e}_I:= \mathbf {e}_{i_1} \wedge \mathbf {e}_{i_2} \cdots \wedge \mathbf {e}_{i_j} \in \bigwedge ^j(K_S^{n+1}).$ Then, the standard basis of $\bigwedge ^j(K_S^{n+1})$ is given by

$$ \begin{align*} \{ \mathbf{e}_I : I=\{i_1,i_2,\ldots,i_j\} \subset \{0,1, \ldots,n \} \,\, \text{with} \,\, i_1 < i_2 < \cdots < i_j \},\end{align*} $$

and for any element $\mathbf {b}= \sum\limits _{\scriptsize {\begin {array}{rcl}I\subseteq \{0,1,\ldots ,n\},\\ \# I=j\end {array}}} b_I \mathbf {e}_I,$ we define $\|\mathbf {b}\| = \displaystyle \max _{I} \|b_I\|.$

Hence, we can write the earlier mentioned $\mathbf {w} \in \bigwedge ^j (\mathcal {O}_S^{n+1})$ as $\mathbf {w}= \sum _{I} w_I \mathbf {e}_I,$ where $w_I \in \mathcal {O}_S.$ We now find the expression of $ (g_{\varepsilon _v,\delta _v})_{v} (g_{\mathbf {f}^{(v)}(x_v)})_{v} \mathbf {w}$ componentwise. Note that $g_{\mathbf {f}^{(v)}(x_v)}$ leaves $\mathbf {e}_{0}^{(v)}$ invariant and sends $\mathbf {e}_{i}^{(v)}$ to $f_{i}^{(v)}(x_v) \mathbf {e}_{0}^{(v)} + \mathbf {e}_{i}^{(v)}$ for all $i=1,\ldots ,n.$ Therefore,

$$ \begin{align*} (g_{\mathbf{f}^{(v)}(x_v)}) (\mathbf{e}_{I}^{(v)} ) = \left \{ \begin{array}{@{}ll}\displaystyle \mathbf{e}_{I}^{(v)} & \text{if} \,\, 0\in I, \\ \displaystyle \mathbf{e}_{I}^{(v)} + \sum_{i\in I} \pm f_{i}^{(v)}(x_v) \mathbf{e}_{I \cup \{0\} \setminus \{i\}}^{(v)} &\text{otherwise} , \end{array} \right.\end{align*} $$

and

$$ \begin{align*} \begin{aligned} & (g_{\varepsilon_v,\delta_v}) (g_{\mathbf{f}^{(v)}(x_v)}) (\mathbf{e}_{I}^{(v)})\\ &\qquad = \left \{ \begin{array}{@{}ll}\displaystyle {\varepsilon_{v}^{(1)}}^{-1} \bigg(\prod_{i \in I \setminus \{0\}} {\delta_{v}^{(i)}}^{-1} \bigg) \mathbf{e}_{I}^{(v)} & \text{if} \,\, 0\in I ,\\ \displaystyle \bigg(\prod_{i \in I} {\delta_{v}^{(i)}}^{-1}\bigg) \mathbf{e}_{I}^{(v)} \pm {\varepsilon_{v}^{(1)}}^{-1} \sum_{i\in I} \bigg(\prod_{r \in I \setminus \{i\}} {\delta_{v}^{(r)}}^{-1} \bigg) f_{i}^{(v)}(x_v) \mathbf{e}_{I \cup \{0\} \setminus \{i\}}^{(v)} &\text{otherwise}. \end{array} \right. \end{aligned} \end{align*} $$

Thus, $(g_{\varepsilon _v,\delta _v})_{v} (g_{\mathbf {f}^{(v)}(x_v)})_{v} \mathbf {w}$ is given by

(5-2) $$ \begin{align} ((g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \mathbf{w})^{(v)}&= \displaystyle \sum_{0 \notin I} \bigg(\prod_{i \in I} {\delta_{v}^{(i)}}^{-1} \bigg) w_{I} \mathbf{e}_{I}^{(v)} + {\varepsilon_{v}^{(1)}}^{-1} \sum_{0 \in I } \bigg( \bigg(\prod_{i \in I \setminus \{0\}} {\delta_{v}^{(i)}}^{-1} \bigg) w_{I} \nonumber\\& \qquad + \displaystyle \sum_{i\notin I} \pm \bigg(\prod_{r \in I \cup \{i\} \setminus \{0\}} {\delta_{v}^{(r)}}^{-1} \bigg) f_{i}^{(v)}(x_v) w_{I \cup \{i\} \setminus \{0\}} \bigg) \mathbf{e}_{I}^{(v)}. \end{align} $$

We now check the conditions (C1) and (C2).

$\bullet $ Checking Condition (C1). Due to (5-2), we can form an explicit expression for $\text {cov} (h(\mathbf {x}) \Delta )$ and that is given by

$$ \begin{align*}\text{cov} (h(\mathbf{x}) \Delta)= a_{K} c ((g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \mathbf{w}) = a_K \prod_v \bigg\|\bigg((g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \mathbf{w}\bigg)^{(v)} \bigg\|_{v,2}.\end{align*} $$

In view of (5-2), it is easy to observe that the components of all the coordinates of

$$ \begin{align*}((g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \mathbf{w})^{(v)}\end{align*} $$

are linear combinations of $1,f_{1}^{(v)}, \ldots , f_{n}^{(v)}.$ Then, Condition (C1) immediately follows from Hypothesis $(\text{i})$ of Proposition 4.3, Lemma 3.9(b,c) and Lemma 3.10. So, there exists $C, \alpha>0$ such that $ \mathbf {x} \mapsto \text {cov} (h(\mathbf {x}) \Delta ) $ is $(C,\alpha )$ -good on $\tilde {B}$ with respect to $\mu ,$ where $C$ and $\alpha $ are explicitly computed in terms of $ C_{v_1}, \ldots , C_{v_{\ell }}, \alpha _{v_1}, \ldots , \alpha _{v_{\ell }}.$

$\bullet $ Checking Condition (C2). By the compactness of the unit sphere in $K_v^{n+1}$ and Hypothesis $(\text{ii})$ of Proposition 4.3, there exists $\rho _v>0$ such that for any $\mathbf {a}=(a_0,a_1,\ldots ,a_n) \in K_v^{n+1}$ ,

(5-3) $$ \begin{align} \displaystyle \sup_{\tiny x_v \in B_v\,\cap \,\text{supp}\,(\mu_v)} |a_0 + a_1 f_{1}^{(v)}(x_v) + \cdots +a_n f_{n}^{(v)}(x_v) |_v \geq \rho_v \|\mathbf{a}\|_{v,2}, \end{align} $$

where $B=\prod _{v\in S}B_v.$

Using (5-2) and (5-3),

$$ \begin{align*} \displaystyle \sup_{\tiny x_v \in B_v\,\cap \,\text{supp}\,(\mu_v)} \bigg\|\bigg((g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \mathbf{w}\bigg)^{(v)} \bigg\|_{v,2} \geq \rho_v |\varepsilon_v^{(1)}|_{v}^{-1} \bigg(\prod_{i=1}^{n} |\delta_{v}^{(i)}|_{v}^{-1}\bigg) \max_I |w_I|_{v} , \end{align*} $$

since $|\delta _{v}^{(i)}|_{v} \geq 1$ for all $i=1,\ldots ,n$ and each $v \in S.$ Therefore,

$$ \begin{align*} \displaystyle \sup_{\tiny x \in B\,\cap \,\text{supp}\,(\mu)} c \bigg((g_{\varepsilon_v,\delta_v})_{v} (g_{\mathbf{f}^{(v)}(x_v)})_{v} \mathbf{w}\bigg) & \geq \displaystyle \prod_{v\in S} \bigg( \rho_v |\varepsilon_v^{(1)}|_{v}^{-1} \bigg(\prod_{i=1}^{n} |\delta_{v}^{(i)}|_{v}^{-1} \bigg) \max_I |w_I|_{v} \bigg) \nonumber \\ &\displaystyle \geq \bigg( \prod_{v \in S} |\varepsilon_{v}^{(1)}|_{v} |\delta_{v}^{(1)}|_{v} \cdots |\delta_{v}^{(n)}|_{v} \bigg)^{-1} \max_I c(w_I) \bigg(\prod_{v\in S} \rho_v \bigg) \end{align*} $$

and, hence,

$$ \begin{align*} \|\text{cov}(h(\mathbf{x}))\Delta\|_{\mu,B} \displaystyle \geq (\!\sqrt{D_K})^j \bigg( \prod_{v \in S} |\varepsilon_{v}^{(1)}|_{v} |\delta_{v}^{(1)}|_{v} \cdots |\delta_{v}^{(n)}|_{v}\bigg)^{-1} \bigg(\prod_{v\in S} \rho_v\bigg), \end{align*} $$

since $a_{K} \geq (\!\sqrt {D_K})^j $ and $w_I \in \mathcal O_S$ for each $I.$

Let

$$ \begin{align*} \displaystyle \tilde{\rho}\ & {\stackrel{\text{def}}{=}}\ (\!\sqrt{D_K})^j \bigg( \prod_{v \in S} |\varepsilon_{v}^{(1)}|_{v} |\delta_{v}^{(1)}|_{v} \cdots |\delta_{v}^{(n)}|_{v}\bigg)^{-1} \bigg(\prod_{v\in S} \rho_v\bigg) \nonumber \\ & \displaystyle = (\!\sqrt{D_K})^j (\text{const}_K)^{-(n+1)} \bigg(\prod_{v\in S} \rho_v \bigg). \end{align*} $$

So, Condition (C2) holds with

(5-4) $$ \begin{align} \rho = \min \{1,\tilde{\rho} \}. \end{align} $$

As a consequence of Theorem 5.1, Proposition 4.3 follows immediately with

(5-5) $$ \begin{align} \tilde{C}= C (N_X D^2)^{n+1} (n+1)^{1+(({\alpha |S_r|})/{2})+\alpha |S_c| } \bigg(\frac{\sqrt{D_K}}{\rho}\bigg)^{\alpha}. \end{align} $$

$\Box $