Diophantine inheritance for p-adic measures

TL;DR

This paper establishes $p$-adic analogues of Kleinbock's theorems on Diophantine exponents inheritance for affine subspaces, introducing a new $p$-adic exponent linked to homogeneous dynamics.

Contribution

It proves complete $p$-adic versions of Kleinbock's theorems and confirms a conjecture of Kleinbock and Tomanov, introducing a novel $p$-adic Diophantine exponent.

Findings

Confirmed a conjecture of Kleinbock and Tomanov.

Established $p$-adic analogues of inheritance theorems.

Introduced a new $p$-adic Diophantine exponent.

Abstract

In this paper we prove complete $p$-adic analogues of Kleinbock's theorems \cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine exponents for affine subspaces. In particular, we answer in the affirmative (and in a stronger form), a conjecture of Kleinbock and Tomanov \cite{KT}, as well as a question of Kleinbock \cite{Kleinbock-exponent}. Our main innovation is the introduction of a new $p$-adic Diophantine exponent which is better suited to homogeneous dynamics, and which we show to be closely related to the exponent considered by Kleinbock and Tomanov.

Equations

$$|q_0+\mathbf{q}\cdot \mathbf{y}{|}_p\le \Vert \tilde{\mathbf{q}}{\Vert }_{\infty }^{-v}.$$

$$(x,x^2,\dots ,x^n)\subset {\mathbb{R}}^n$$

$$w(\mu )=\sup \{v:\mu (\{y|w(y)>v\})>0\}.$$

$$w(\mathcal{M})=w(\mathcal{L})=\inf \{w(\mathbf{y})|\mathbf{y}\in \mathcal{L}\}=\inf \{w(\mathbf{y})|\mathbf{y}\in \mathcal{M}\}.$$

$$w_p(\mu ):=\sup \{v:\mu (\{y|w_p(y)>v\})>0\}$$

$$w_p(\mathcal{M})=w_p(\mathcal{L})=\inf \{w_p(\mathbf{y})|\mathbf{y}\in \mathcal{L}\}=\inf \{w_p(\mathbf{y})|\mathbf{y}\in \mathcal{M}\}.$$

$$w_p(\mathcal{L})=\max (n,w_p(A))\text{ and }w(\mathcal{L})=\max (n+1,w(A)).$$

$$\forall x\in A\text{ is a center of some ball of }B,$$

$$1_A\le \sum _i{1}_{B_i}\le N_X.$$

$$\underset{\begin{array}{c}x\in \mathrm{supp}\mu ,r>0\\ B(x,3r)\subset U\end{array}}{\sup }\frac{\mu (B(x,3r))}{\mu (x,r)}<D.$$

$$\mathcal{L}=⟨f(B\cap \mathrm{supp}\mu ){⟩}_a\forall \text{ nonempty open }B\text{ with }\mu (B)>0.$$

$$\mu\left(\{x\in B\big{|}|f(x)|<\varepsilon\}\right)\leq C\left(\displaystyle\frac{\varepsilon}{\sup_{x\in B}|f(x)|}\right)^{\alpha}|B|.$$

$$w({\mathbf{f}}_{\ast }\mu )=w(\mathcal{L})=\inf \{w(\mathbf{y})|\mathbf{y}\in \mathcal{L}\}=\inf \{w(\mathbf{f}(x))|x\in \mathrm{supp}\mu \}.$$

$$|q_0+\mathbf{q}.\mathbf{y}{|}_p$$

$$\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }^{n+1}$$

$$\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }.|q_0+\mathbf{q}.\mathbf{y}{|}_p$$

$$\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }$$

$$\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }.|q_0+\mathbf{q}.\mathbf{y}{|}_p$$

$$\Vert \mathbf{q}{\Vert }_p\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }$$

$$|\mathbf{q}.\mathbf{y}+q_0{|}_p<\frac{1}{(||\mathbf{q}|{|}_p||\tilde{\mathbf{q}}|{|}_{\infty }{)}^v||\tilde{\mathbf{q}}|{|}_{\infty }}.$$

$$w_p(\mathbf{y}):=\sup \{v\text{ appearing in }(\text{ref}{Z}_S)\}.$$

$$w(\mathbf{y}):=\sup \{v\text{ appearing in }(\text{ref}def:exp)\}.$$

$$E=\left\{|q_{0}+\mathbf{q}.\mathbf{y}|_{p}\|\tilde{\mathbf{q}}\|_{\infty},\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}~{}\big{|}~{}\tilde{\mathbf{q}}=(q_{0},\mathbf{q})\in\mathbb{Z}[1/p]^{n+1}\right\}.$$

$$|q_{0k}+{\mathbf{q}}_k.\mathbf{y}{|}_p\Vert {\tilde{\mathbf{q}}}_k{\Vert }_{\infty }\to 0,\text{ as }k\to \infty ,$$

$$|q_{0k}{|}_p\Vert {\tilde{\mathbf{q}}}_k{\Vert }_{\infty }\le |q_{0k}+{\mathbf{q}}_k.\mathbf{y}{|}_p\Vert {\tilde{\mathbf{q}}}_k{\Vert }_{\infty }+\Vert {\mathbf{q}}_k{\Vert }_p\Vert {\tilde{\mathbf{q}}}_k{\Vert }_{\infty }\Vert \mathbf{y}{\Vert }_p,$$

$$|q_{ik}{|}_p|q_{ik}{|}_{\infty }=p^{-m_{ik}}.p^{m_{ik}}|z_{ik}{|}_{\infty }\le \Vert {\tilde{\mathbf{q}}}_k{\Vert }_p\Vert {\tilde{\mathbf{q}}}_k{\Vert }_{\infty }\le M.$$

$$p^{-m_{ik}}.p^{m_k}|z_k{|}_{\infty }\le \Vert {\tilde{\mathbf{q}}}_k{\Vert }_p\Vert {\tilde{\mathbf{q}}}_k{\Vert }_{\infty }\le M,$$

$$p^{-m_k}.p^{m_{ik}}|z_{ik}{|}_{\infty }\le \Vert {\tilde{\mathbf{q}}}_k{\Vert }_p\Vert {\tilde{\mathbf{q}}}_k{\Vert }_{\infty }\le M.$$

$$w_p(\mathbf{y})=w(\mathbf{y})+1.$$

$$|\mathbf{q}.\mathbf{y}+q_0{|}_p<\frac{1}{(\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }{)}^v}.$$

$$|\mathbf{q}.\mathbf{y}+q_0{|}_p<\frac{1}{(\Vert \mathbf{q}{\Vert }_p\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }{)}^{v-1}\Vert \tilde{\mathbf{q}}{\Vert }_{\infty }}.$$

$$|\mathbf{q}.\mathbf{y}+q_0{|}_p<\frac{1}{(||\mathbf{q}|{|}_p||\tilde{\mathbf{q}}|{|}_{\infty }{)}^v||\tilde{\mathbf{q}}|{|}_{\infty }},$$

Full text

A. G. gratefully acknowledges support from a grant from the Indo-French Centre for the Promotion of Advanced Research; a Department of Science and Technology, Government of India Swarnajayanti fellowship and a MATRICS grant from the Science and Engineering Research Board.

Mathematics Subject Classification (2010). Primary 11J83; Secondary 11J54, 11J61, 37A45.

1. Introduction

This paper is concerned with the study of $p$-adic Diophantine approximation on manifolds. We briefly recall the setting and basic results from the paper [23] of Kleinbock and Tomanov.

For $\mathbf{q}=(q_{1},\dots,q_{n})\in\mathbb{Z}^{n}$ and $q_{0}\in\mathbb{Z}$, set $\tilde{\mathbf{q}}:=(q_{0},q_{1},\dots,q_{n})$. Define the Diophantine exponent $w(\mathbf{y})$ of $\mathbf{y}\in\mathbb{Q}_{p}^{n}$ to be the supremum of $v>0$ such that there are infinitely many $\tilde{\mathbf{q}}\in\mathbb{Z}^{n+1}$ such that

[TABLE]

In view of Dirichlet’s theorem ([23] §11.2), $w(\mathbf{y})\geq n+1$ for every $\mathbf{y}\in\mathbb{Q}_{p}^{n},$ with equality for Haar almost every $\mathbf{y}$ by the Borel-Cantelli lemma.

A vector $\mathbf{y}\in\mathbb{Q}^{n}_{p}$ is called very well approximable (VWA) if $w(\mathbf{y})>n+1$. The set of very well approximable vectors has zero Haar measure. Diophantine approximation on manifolds or “Diophantine approximation with dependent quantities” is concerned with the question of inheritance of Diophantine properties which are generic with respect to Lebesgue measure in $\mathbb{R}^{n}$ or $\mathbb{Q}_{p}^{n}$ with respect to Lebesgue measure, by appropriate proper subsets. The theory began with a conjecture of Mahler, that almost every point on the Veronese curve

[TABLE]

is not very well approximable. Mahler made his conjecture in the context of a classification of numbers in terms of their approximation properties. We refer the reader to Bugeaud’s book [5] for details about developments in this area. Mahler’s conjecture was settled by Sprindžhuk, in the real, $p$-adic and positive characteristic (i.e. function fields in one variable over a finite field) contexts. It was conjectured by Sprindžuk in 1980 [30, 31] and proved by Kleinbock and Margulis [22] that real analytic manifolds not contained in any proper affine subspace of $\mathbb{R}^{n}$ are extremal. They introduced methods from the ergodic theory of group actions into the subject and [22] has become a cornerstone of the subject. A further breakthrough was achieved by Kleinbock, Lindenstrauss and Weiss in [21], where a wide class of measures, including fractal measures as well as pushforwards of Lebesgue measure by nondegenerate maps which were previously introduced by Kleinbock and Margulis (the definition follows in the next paragraph) were studied in a unified manner. Subsequently, Kleinbock and Tomanov [23] proved an $S$-adic version of Sprindžhuk’s conjectures and in [12], the second named author proved the positive characteristic version of Sprindžhuk’s conjecture. In two important papers, [19, 20], D. Kleinbock systematically explored Diophantine approximation on affine subspaces and their nondegenerate submanifolds and the second named author proved Khintchine type theorems both convergence and divergence [10, 11, 13, 14, 15]. Inhomogeneous and quantitative versions of Khintchine’s theorem for affine subspaces have been proved in [3, 9]. We refer the reader to the survey [16] of the second named author for more details on the problem of Diophantine approximation on affine subspaces.

The subject of $p$-adic Diophantine approximation started with the work of E. Lutz [24] and has seen numerous advances in different contexts over the years including early work of Mahler [25, 26]. We refer the reader to [5] for a comprehensive reference, to [23] and the references therein for work relating to $p$-adic metric Diophantine approximation on manifolds, and to [4, 1, 6, 8] for recent results.

In this paper, we provide a complete $p$-adic analogue of the results of D. Kleinbock [20] on Diophantine exponents of affine subspaces in two types of approximations, one was considered by Kleinbock and Tomanov in [23] and the other one is a new type of approximation we introduced here. This answers, in a stronger form, Conjecture IS of Kleinbock and Tomanov [23] as well as a question of D. Kleinbock [20].

While the methods of the present paper are heavily influenced by the work of D. Kleinbock, providing a complete $p$-adic analogue of his results poses substantial new difficulties. To circumvent these difficulties, one of the key innovations of the present paper is a new $p$-adic Diophantine exponent which measures approximation with respect to $\mathbb{Z}[1/p]$-points.

We now introduce some notation and state the main results in the present paper. For a Borel measure $\mu$ on $\mathbb{Q}_{p}^{n}$, we follow [20] in defining the Diophantine exponent $\omega(\mu)$ of $\mu$ to be

[TABLE]

The exponent only depends on the measure class of $\mu$. Let $\lambda$ denote Haar measure on $\mathbb{Q}^{n}_{p}$ normalized so that $\mathbb{Z}^{n}_{p}$ has volume $1$, the dimension being clear from the context. If $\mathcal{M}\subset\mathbb{Q}_{p}^{n}$ is a $d$ dimensional analytic manifold, and $\mu$ is the pushforward of Haar measure on $\mathbb{Q}_{p}^{d}$ by a map parametrising $\mathcal{M}$, then $\omega(\mathcal{M})$ is defined to be $\omega(\mu)$. Following Kleinbock [20], we say that a differentiable map $f:U\to\mathbb{Q}^{n}_{p}$, where $U$ is an open subset of $\mathbb{Q}^{d}_{p}$, is nondegenerate in an affine subspace $\mathcal{L}$ of $\mathbb{Q}^{n}_{p}$ at $x\in U$ if $f(U)\subset\mathcal{L}$ and the span of all the partial derivatives of $f$ at $x$ up to some order coincides with the linear part of $\mathcal{L}$. If $\mathcal{M}$ is a $d$-dimensional submanifold of $\mathcal{L}$, we will say that $\mathcal{M}$ is nondegenerate in $\mathcal{L}$ at $y\in\mathcal{M}$ if there exists a diffeomorphism $f$ between an open subset $U$ of $\mathbb{Q}^{d}_{p}$ and a neighbourhood of $y$ in $\mathcal{M}$ is nondegenerate in $\mathcal{L}$ at $f^{-1}(y)$. Finally, we will say that $f:U\to\mathcal{L}$ (resp., $\mathcal{M}\subset\mathcal{L}$) is nondegenerate in $\mathcal{L}$ if it is nondegenerate in $\mathcal{L}$ at $\lambda$-a.e. point of $U$ (resp., of $\mathcal{M}$, in the sense of the smooth measure class on $\mathcal{M}$). Here is a special case of one of our main results.

Theorem 1.1**.**

Let $\mathcal{L}$ be an affine subspace of $\mathbb{Q}_{p}^{n}$, and let $\mathcal{M}$ be a submanifold of $\mathcal{L}$ which is nondegenerate in $\mathcal{L}$. Then

[TABLE]

In particular, this implies that if $\mathcal{L}$ is an extremal subspace of $\mathbb{Q}_{p}^{n}$, and $\mathcal{M}$ is nondegenerate in $\mathcal{L}$, then $\mathcal{M}$ is extremal. As noted by Kleinbock in the context of real Diophantine approximation, the middle equality is non-obvious and non-trivial. The difficulty of proving it persists in the $p$-adic context and is in fact compounded by the difficulty, in the $p$-adic setting, of translating the Diophantine problem to dynamics and back. To overcome this challenge, we introduce the new idea of $\mathbb{Z}[1/p]$-exponents. We define these exponents $w_{p}(\mathbf{y})$ and $w_{p}(A)$ in the next sections; the point is that these exponents seem to be better suited to techniques from homogeneous dynamics and in particular allow us to prove an “if and only if” Dani-type correspondence in the $p$-adic setting, which does not seem to be readily achieved whilst dealing with approximation of $p$-adic vectors by rational numbers. We suspect these exponents will find further use. We then explain how the $\mathbb{Z}[1/p]$-exponent is related to the usual exponent. We also define $p$-adic exponents $\omega_{p}(\mu)$ of a borel measure $\mu$ on $\mathbb{Q}_{p}^{n}$ to be

[TABLE]

and prove the following theorem.

Theorem 1.2**.**

Let $\mathcal{L}$ be an affine subspace of $\mathbb{Q}_{p}^{n}$, and let $\mathcal{M}$ be a submanifold of $\mathcal{L}$ which is nondegenerate in $\mathcal{L}$. Then

[TABLE]

We will also provide a condition for $p$-adic extremality in terms of Diophantine properties of the parametrising matrix of an affine subspace.

Theorem 1.3**.**

Let $\mathcal{L}$ be an affine subspace, parametrized by a matrix $R_{A}$ as in (7.1). If all the rows (resp. columns) are rational multiples of one row (resp. column) then one has

[TABLE]

Diophantine approximation on affine subspaces plays in role in KAM theory, see [28, 32] and we hope that the present paper might also find applications. The setup of Diophantine approximation by $\mathbb{Z}[1/p]$-points has already been considered in considerable detail in the context of intrinsic Diophantine approximation on varieties by Ghosh, Gorodnik and Nevo [17, 18]. However, as we have mentioned, as far as we are aware the present paper is the first one to consider it in the context of (extrinsic) Diophantine approximation on manifolds. Moreover, we are not aware of any such connection between exponents of $\mathbb{Z}$-approximation and $\mathbb{Z}[\frac{1}{p}]$-approximation in case of reals.

Structure of the paper

The main results of the paper are proved in the more general setting of Federer measures and nonplanar maps. In the next section we present these and more definitions leading up to the statement of Theorem 2.2 from which Theorem 1.1 follows. Section 3 introduces a $p$-adic exponent $w_{p}$ and discusses the relationship between $w_{p}$ and $w$. In §4 we prove an important dictionary in $p$-adic dynamics: namely the explicit connection between $p$-adic exponents and homogeneous dynamics. In §5 we extend the quantitative nondivergence theorem proved by Kleinbock to the $p$-adic setting. The subsequent section 6 deals with applications on nondivergence to $p$-adic exponents, in particular proving Theorem 2.2. The final section 7 deals with higher Diophantine exponents.

Acknowledgements

Part of this work was done when both authors were visiting the Weizmann institute. We thank Uri Bader and the Weizmann institute for the hospitality. A. G. gratefully acknowledges the support of the Benoziyo Endowment Fund for the Advancement of Science at the Weizmann Institute. Another part of this work was done when the authors were visiting the ICTS, Bengaluru. We thank the institute for the pleasant working conditions.

2. Preliminaries

Measures and spaces

A metric space $X$ is called Besicovitch [23] if there exists a constant $N_{X}$ such that the following holds: for any bounded subset $A$ of $X$ and for any family $\mathcal{B}$ of nonempty open balls in $X$ such that

[TABLE]

there is a finite or countable subfamily $\{B_{i}\}$ of $B$ with

[TABLE]

As remarked in [23], any separable ultrametric space $X$ is Besicovitch with $N_{X}=1$. We now define $D$-Federer measures following [21]. Let $\mu$ be a Radon measure on $X$, and $U$ an open subset of $X$ with $\mu(U)>0$. We say that $\mu$ is $D$-Federer on $U$ if

[TABLE]

Finally, we say that $\mu$ as above is Federer if for $\mu$-a.e. $x\in X$ there exists a neighbourhood $U$ of $x$ and $D>0$ such that $\mu$ is $D$-Federer on $U$. We refer the reader to [21, 23] for examples of Federer measures.

Following, [20], for a subset $M$ of $\mathbb{Q}_{p}^{n}$, define its affine span $\langle M\rangle_{a}$ to be the intersection of all affine subspaces of $\mathbb{Q}_{p}^{n}$ containing $M$. Let $X$ be a metric space, $\mu$ a measure on $X$, $\mathcal{L}$ an affine subspace of $\mathbb{Q}_{p}^{n}$ and $f$ a map from $X$ into $\mathcal{L}$. Say that $(f,\mu)$ is nonplanar in $\mathcal{L}$ if

[TABLE]

$(C,\alpha)$-good functions

In this section, we recall the notion of $(C,\alpha)$-good functions on ultrametric spaces. We follow the treatment of Kleinbock and Tomanov [23]. Let $X$ be a metric space, $\mu$ a Borel measure on $X$ and let $(F,|\cdot|)$ be a valued field. For a subset $U$ of $X$ and $C,\alpha>0$, say that a Borel measurable function $f:U\to F$ is $(C,\alpha)$-good on $U$ with respect to $\mu$ if for any open ball $B\subset U$ centered in $\operatorname{supp}\mu$ and $\varepsilon>0$ one has

[TABLE]

Where $\|f\|_{\mu,B}=\sup\{c:\mu(\{x\in B:|f(x)|>c\})>0\}$.

Theorem 2.1**.**

[Theorem $4.3$ [23]] Let $F$ be either $\mathbb{R}$ or an ultrametric valued field, and let $f$ be a $C^{l}$ map from an open subset $U$ of $F^{d}$ to $F^{n}$. Then $f$ is nonplanar and good at every point of $U$ where it is nondegenerate.

As a corollary, we have

Corollary 2.1**.**

Let $\mathcal{L}$ be an affine subspace of $\mathbb{Q}_{p}^{n}$ and let $f=(f_{1},\dots,f_{n})$ be a smooth map from an open subset $U$ of $\mathbb{Q}_{p}^{d}$ to $\mathcal{L}$ which is nondegenerate in $\mathcal{L}$ at $x_{0}\in U$. Then $f$ is good at $x_{0}$.

Proof.

This follows from Theorem 2.1, see Corollary $3.2$ in [19]. ∎

Main Theorem

Our main result is a complete $p$-adic analogue of Theorem $0.3$ of [20].

Theorem 2.2**.**

Let $\mu$ be a Federer measure on a Besicovitch metric space $X,\mathcal{L}$ an affine subspace of $\mathbb{Q}_{p}^{n}$, and let $\mathbf{f}:X\to\mathcal{L}$ be a continuous map such that $(\mathbf{f},\mu)$ is good and nonplanar in $\mathcal{L}$. Then

[TABLE]

3. $p$-adic Diophantine exponents

We begin with some motivation for $p$-adic Diophantine approximation and the definition of $v$-approximable numbers in $\mathbb{Q}_{p}^{n},$ both using $\mathbb{Z}$ and $\mathbb{Z}[1/p]$ approximations. A natural starting point should be an analogue of Dirichlet’s theorem in this set up. In [23] a $p$-adic Dirichlet’s theorem using $\mathbb{Z}$ approximations has been discussed in detail. Here we observe that Dirichlet’s theorem using $\mathbb{Z}$ approximations does indeed give a Dirichlet theorem in case of $\mathbb{Z}[1/p]$ approximations.

Here and below, we adopt the notation $\tilde{\mathbf{q}}:=(q_{0},\mathbf{q})$ for $q_{0},\mathbf{q}$ in $\mathbb{Z}$ as well as $\mathbb{Z}[1/p]$. Lets recall the Dirichlet’s theorem in [23], for $\mathbf{y}\in\mathbb{Q}_{p}^{n}$ and for every $Q>0$ there exists an integer solution for the system

[TABLE]

Now observe that this implies that for every $Q>0$ there exists an integer solution to the system

[TABLE]

Note that $\|{\mathbf{q}}\|_{p}\leq 1$ for integer vectors, so we have that for any $\mathbf{y}\in\mathbb{Q}_{p}^{n}$ and every $Q>0$ there exists a $\mathbb{Z}[1/p]^{n+1}$ solution to the system

[TABLE]

This motivates the follwoing definition.

Definition 3.1**.**

v-$\mathbb{Z}[1/p]$ approximable vectors: $\mathbf{y}\in\mathbb{Q}_{p}^{n}$ is $v-\mathbb{Z}[1/p]$-approximable if there exist $\tilde{\mathbf{q}}=(q_{0},\mathbf{q})\in\mathbb{Z}[1/p]^{n+1}$ with unbounded $||\mathbf{q}||_{p}||\tilde{\mathbf{q}}||_{\infty}$ such that

[TABLE]

We will denote $v-\mathbb{Z}[1/p]$-approximable points by $\mathcal{W}_{v}^{p}$ and also define

[TABLE]

Similarly, we will denote $v$-approximable points by $\mathcal{W}_{v}$ and recall that we have defined

[TABLE]

We therefore have two Diophantine exponents:

1. (1)

$\omega$, defined in (1.1) involving $\mathbb{Z}$-Diophantine approximation of $p$-adic vectors. 2. (2)

$\omega_{p}$, defined in (3.8) involving $\mathbb{Z}[1/p]$-Diophantine approximation of $p$-adic vectors.

Later on, we will need higher Diophantine exponents of both kinds. Although the two types of approximations are a priori different, we will shortly prove that the Diophantine exponents are very closely related. The following lemma will come handy to compare these Diophantine exponents and further to relate Diophantine approximation to dynamics.

Lemma 3.1**.**

Consider the set

[TABLE]

If $(x_{k},z_{k})\in E$ such that $z_{k}$ is bounded and $x_{k}\to 0$ then $x_{k}=0$ for all but finitely many $k$.

Proof.

Suppose $\|\mathbf{q}_{k}\|_{p}\|\tilde{\mathbf{q}}_{k}\|_{\infty}\leq M$ for some $M>0$ and

[TABLE]

where $\tilde{\mathbf{q}}_{k}=(q_{0k},\mathbf{q}_{k})=(q_{ik})_{i=0}^{n}\in\mathbb{Z}[1/p]^{n+1}$. Since

[TABLE]

we can choose $M$ such that $|q_{0k}|_{p}\|\tilde{\mathbf{q}}_{k}\|_{\infty}\leq M$ and $\|\mathbf{q}_{k}\|_{p}\|\tilde{\mathbf{q}}_{k}\|_{\infty}\leq M$ and therefore $\|\tilde{\mathbf{q}}_{k}\|_{p}\|\tilde{\mathbf{q}}_{k}\|_{\infty}\leq M$.

Note that there are only finitely many $p$-free integers in $q_{ik},$ i.e. in $\tilde{\mathbf{q}}_{k}=(q_{ik})_{i=0}^{n}=(p^{m_{ik}}z_{ik})_{i=0}^{n}$ where $p\nmid z_{ik},$ $|z_{ik}|_{\infty}$ is bounded. This follows from the fact that

[TABLE]

So there are finitely many $z_{ik}$.

We denote $\|\tilde{\mathbf{q}}_{k}\|_{\infty}=p^{m_{k}}z_{k}$ where $p\nmid z_{k},m_{k}\in\mathbb{Z}$. If $z_{k}=0$ then $\tilde{\mathbf{q}}_{k}=0$. Otherwise

[TABLE]

and similarly,

[TABLE]

So for nonzero elements $q_{ik}=p^{m_{ik}}z_{ik}$ we have $|m_{k}-m_{ik}|$ bounded since we have already noted that there are finitely many $z_{ik}$. Therefore $|q_{0k}+\mathbf{q}_{k}.\mathbf{y}|_{p}\|\tilde{\mathbf{q}}_{k}\|_{\infty}$ has only finitely many options. So it can only go to [math] if the terms are identically [math] for all but finitely many possibilities. ∎

Now we can conclude the following relations between exponents.

Proposition 3.1**.**

For any $\mathbf{y}\in\mathbb{Q}_{p}^{n}$ we have

[TABLE]

Proof.

Suppose $\mathbf{y}\in\mathcal{W}_{v},$ then there exists infinitely many integer vectors $\tilde{\mathbf{q}}=(q_{0},\mathbf{q})\in\mathbb{Z}^{n+1}$ such that

[TABLE]

Since $\|\mathbf{q}\|_{p}\leq 1$ for ${\mathbf{q}}\in\mathbb{Z}^{n+1}$, the above inequality is the same as

[TABLE]

Lemma 3.1 assures us that the $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$ appearing here are unbounded when $v>1$. Hence, we have that $\mathcal{W}_{v}\subset\mathcal{W}_{v-1}^{p}$ when $v>1$. On the other hand, if $\mathbf{y}\in\mathcal{W}_{v}^{p}$ then there are unbounded many $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$ such that

[TABLE]

where $\tilde{\mathbf{q}}=(q_{0},\mathbf{q})\in\mathbb{Z}[1/p]^{n+1}$. This inequality can be rewritten as

[TABLE]

for unbounded many $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$ where $\tilde{\mathbf{q}}=(q_{0},\mathbf{q})\in\mathbb{Z}[1/p]^{n+1}$. But note that $\|\mathbf{q}\|_{p}\mathbf{q}\in\mathbb{Z}^{n+1}$. Hence the above criterion is the same as

[TABLE]

for unbounded many $\|\tilde{\mathbf{q}}\|_{\infty}$ where $\tilde{\mathbf{q}}=(q_{0},\mathbf{q})\in\mathbb{Z}[1/p]\times\mathbb{Z}^{n}$. Now note that $|q_{0}|_{p}\leq\max(\|\mathbf{y}\|_{p},1)$. So when $q_{0}\notin\mathbb{Z}$, we may multiply by $|q_{0}|_{p}>1$ and since we have an upper bound for $|q_{0}|_{p}$ we get a $\mathbb{Z}$ approximation but for slightly smaller $v$. So we have for any $\varepsilon>0$

[TABLE]

for unbounded many $\|\tilde{\mathbf{q}}\|_{\infty},$ and hence, for infinitely many $\tilde{\mathbf{q}}\in\mathbb{Z}^{n+1}$. Therefore $\mathcal{W}_{v}\subset\mathcal{W}_{v+1-\varepsilon}^{p}$. Hence the conclusion follows. ∎

A quick observation which follows from Dirichlet’s theorem and Proposition 3.1 is that $w_{p}(\mathbf{y})\geq n$ for all $\mathbf{y}\in\mathbb{Q}_{p}^{n}$. We can define the Diophantine $\mathbb{Z}[1/p]$ exponent more generally for a matrix $A$ of order $m,n$ and we will need these notions later in the paper. Define

[TABLE]

and similarly the Diophantine $\mathbb{Z}$ exponent as

[TABLE]

A similar reasoning as before shows that $w_{p}(A)=w(A)+1$.

4. Connecting Diophantine approximation and homogeneous dynamics

We will weaken the hypothesis of Lemma 2.1 of [20] with the same conclusion.

Lemma 4.1**.**

Suppose we are given a set $E\subset\mathbb{R}^{2}$ such that

• •

If the second coordinate of $E$ is bounded then the first coordinate cannot converge to [math] unless it is [math] ultimately, i.e. $(x_{n},z_{n})\in E$ such that $|z_{n}|$ is bounded and $x_{n}\to 0$ implies that $x_{n}=0$ for all but finitely many $n$.

• •

$(0,z)\in E\implies(0,kz)\in E~{}~{}\text{ for infinitely many }k\in\mathbb{N}$.

Take $a,b>0$ and $v>\frac{a}{b}$ and define

[TABLE]

As before, $p$ is a prime. Then the following are equivalent:

1. (1)

There exists $(x,z)\in E$ with arbitrarily large $|z|$ such that

[TABLE] 2. (2)

There exists arbitrarily large $t>0$ such that for some $(x,z)\in E\setminus\{0\}$ one has

[TABLE]

Proof.

We first show that $(1)$ implies $(2)$. There exists $(x,z)\in E$ with arbitrary large $|z|$ such that $|x|\leq|z|^{-v}$. Define $t>0$ such that $p^{-bt}|z|=p^{-ct}$; this is possible since $b-c>0$. Then

[TABLE]

We now show that $(2)$ implies $(1)$. Accordingly, we assume that there exists a sequence of positives $\{t_{n}\}\to\infty$ such that

[TABLE]

Therefore

[TABLE]

If $\{z_{n}\}$ is unbounded then $(1)$ is proved. Suppose then, that $\{z_{n}\}$ is bounded. Since $\{x_{n}\}\to 0$, by the hypothesis $x_{n}=0$ for all but finitely many $n$. Therefore we have that $(0,z_{m})\in E\setminus\{0\}$. By the hypothesis, $(0,kz_{m})\in E\setminus\{0\}$ for inifinitely many $k\in\mathbb{N}$, which will satisfy $(1)$. ∎

Proposition 4.1**.**

For $\mathbf{y}\in\mathbb{Q}_{p}^{n}$, the following are equivalent

1. (1)

$\mathbf{y}\in\mathcal{W}^{p}_{v}$, where $v>{n}$, 2. (2)

there exists arbitrarily large $t>0$ such that

[TABLE]

where $a=\frac{n}{n+1},b=\frac{1}{n+1},c=\frac{v-n}{(n+1)(v+1)}\Leftrightarrow v=\frac{n(1+c)+c}{1-(n+1)c}\text{ and }\tilde{\mathbf{q}}=(q_{0},\mathbf{q})\in\mathbb{Z}[1/p]^{n+1}$.

Proof.

We will apply Lemma 4.1 to the set

[TABLE]

By Lemma 3.1, this set satisfies the first hypothesis of Lemma 4.1. Suppose

[TABLE]

then $|q_{0}+\mathbf{q}.\mathbf{y}|_{p}\|\tilde{\mathbf{q}}\|_{\infty}=0$. For any $u\in\mathbb{N}$ such that $p\nmid u$,

[TABLE]

and

[TABLE]

implies that $(0,u\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty})\in E$, giving the second hypothesis of Lemma 4.1. Now this Proposition follows directly from Lemma 4.1. ∎

5. Quantitative Nondivergence for flows on homogeneous spaces

We begin this section by stating Theorem $2.1$ of [20]. This theorem is an improvement of an original theorem of Kleinbock and Margulis ([22]). This improvement was the main tool in D. Kleinbock’s approach to studying Diophantine exponents of subspaces and their nondegenerate submanifolds. We will use a $p$-adic version of the Theorem, which in turn constitutes an improvement of the nondivergence theorem in [23]. Nondivergence estimates for flows on homogeneous spaces have a rich history, we refer the reader to [20] and the references therein.

Theorem 5.1**.**

Let ${k},N\in\mathbb{Z}_{+}$ and $C,\alpha,D>0$. Suppose that we are given an $N$-Besicovitch metric space $X$, a weighted poset $(\mathcal{B},\eta)$, a ball $B=B(x,r)$ in $X$, a measure $\mu$ which is $D$-Federer on \tilde{B}=B\big{(}x,3^{m}r\big{)}, and a mapping $\psi:\mathcal{B}\to C(\tilde{B})$, $s\mapsto\psi_{s}$, such that the following holds:

1. (1)

$\ell(\mathcal{B})\leq{k}$; 2. (2)

$\forall\,s\in\mathcal{B}\,,\quad\psi_{s}$* is $(C,\alpha)$ on $\tilde{B}$ with respect to $\mu$; * 3. (3)

$\forall\,s\in\mathcal{B}\,,\quad\|\psi_{s}\|_{\mu,B}\geq\eta(s)$; 4. (4)

\forall\,y\in\tilde{B}\,\cap\,\operatorname{supp}\mu,\quad\#\{s\in\mathcal{B}\bigm{|}|\psi_{s}(y)|<\eta(s)\}<\infty.

Then $\forall\,\varepsilon>0$ one has

[TABLE]

In the following discussion we assume,

• •

$\mathcal{D}$ is an integral domain, that is, a commutative ring with $1$ and without zero divisors;

• •

$K$ is the quotient field of $\mathcal{D}$;

• •

${\mathcal{R}}$ is a commutative ring containing ${K}$ as a subring.

The following Theorem is an improvement of Theorem 6.3 of [23] using the improved quantitative nondivergence i.e. Theorem 5.1 of D. Kleinbock. We refer the reader to loc.cit. for the definition of norm-like functions.

Theorem 5.2**.**

Let $X$ be a metric space, $\mu$ a uniformly Federer measure on $X$, and let ${\mathcal{D}}\subset{K}\subset{\mathcal{R}}$ be as above, ${\mathcal{R}}$ being a topological ring. For $m\in\mathbb{N}$, let a ball $B=B(x_{0},r_{0})\subset X$ and a continuous map $h:\tilde{B}\to\operatorname{GL}(m,{\mathcal{R}})$ be given, where $\tilde{B}$ stands for $B(x_{0},3^{m}r_{0})$. Also let $\nu$ be a norm-like function on $\mathcal{M}({\mathcal{R}},{\mathcal{D}},m)$. For any $\Delta\in\mathcal{P}({\mathcal{D}},m)$ denote by $\psi_{\Delta}$ the function x\mapsto\nu\big{(}h(x)\Delta\big{)} on $\tilde{B}$. Now suppose for some $C,\alpha>0$ one has

(i)* for every $\Delta\in\mathcal{P}({\mathcal{D}},m)$, the function $\psi_{\Delta}$ is $(C,\alpha)$ on $\tilde{B}$ with respect to $\mu$,*

(ii)* for every $\Delta\in\mathcal{P}({\mathcal{D}},m)$, $\|\psi_{\Delta}\|_{\mu,B}\geq\rho^{\operatorname{rank}\Delta}$,*

(iii)* \forall\,x\in\tilde{B}\,\cap\,\operatorname{supp}\mu,\quad\#\big{\{}\Delta\in\mathcal{P}({\mathcal{D}},m)\bigm{|}\psi_{\Delta}(x)<\rho\big{\}}<\infty.*

Then for any positive $\varepsilon\leq\rho$ one has

[TABLE]

Proof.

Take $\eta(\Delta)=\rho^{rk(\Delta)}$. We want to show that

[TABLE]

Take x\in\Phi\big{(}\frac{\varepsilon}{\rho},{\mathcal{B}}\big{)}\cap B, so there exists a flag $\mathcal{F}_{x}$. Let

[TABLE]

be all the elements of \mathcal{F}_{x}\cup\big{\{}\{0\},{\mathcal{D}}^{m}\big{\}} such that the following conditions are satisfied:

1. (1)

$\frac{\varepsilon}{\rho}\eta(\Delta)\leq|\psi_{\Delta}(x)|\leq\eta(\Delta)\ \ \forall\delta\in\mathcal{F}_{x}.$ 2. (2)

$|\psi_{\Delta}(x)|\geq\eta(\Delta)\ \ \forall\Delta\in\mathcal{B}(\mathcal{F}_{x}).$

Pick any $\gamma\in{\mathcal{D}}^{m}\setminus\{0\}$. Then there exists $i$, $1\leq i\leq l$, such that $\gamma\in\Delta_{i}\setminus\Delta_{i-1}$. Now $\gamma\notin\Delta_{i-1}=\mathcal{R}\Delta_{i-1}\cap\mathcal{D}^{m}$ implies that $\gamma\notin\mathcal{R}\Delta_{i-1}$ since $\Delta_{i-1}$ is primitive, hence $g\gamma\notin g{\mathcal{R}}\Delta_{i-1}={\mathcal{R}}g\Delta_{i-1}$ for any $g\in\operatorname{GL}(m,{\mathcal{R}})$. Therefore, if one defines $\Delta^{\prime}:={\mathcal{D}}\Delta_{i-1}+{\mathcal{D}}\gamma$, in view of (N2) one has

[TABLE]

Further, let $\Delta:={K}\Delta^{\prime}\,\cap\,{\mathcal{D}}^{m}$. It is a primitive submodule containing $\Delta^{\prime}$ and of rank equal to $\operatorname{rank}(\Delta^{\prime})$, so, by (N1),

[TABLE]

Moreover, it is also contained in $\Delta_{i}$ and contains $\Delta_{i-1}$, since

[TABLE]

Therefore it is comparable to any element of $\mathcal{F}_{x}$, i.e. belongs to $\mathcal{F}_{x}\cup\mathcal{P}(\mathcal{F}_{x})$. Then one can use properties (1) and (2) above to deduce that

[TABLE]

and then, in view of (6.5) and (6.6), conclude that

[TABLE]

∎

We recall Lemma 8.1 from [23]. It is proved for an arbitrary, finite set of places $S$ of $\mathbb{Q}$. We only need it for the case $S=\{\infty,p\}$ for a prime $p$.

Lemma 5.1**.**

The function $\nu:\mathcal{M}(\mathbb{Q}_{S},{\mathbb{Z}}_{S},m)\to\mathbb{R}_{+}$ given by $\,\nu(\Delta)=cov(\Delta)$, with $\,cov(\cdot)$ as defined earlier is norm-like, with $C_{\nu}=1$.

As a consequnece of Theorem 5.2 we get the following refined version of Theorem 8.3 of [23]

Theorem 5.3**.**

Let $X$ be a be a Besicovitch metric space, $\mu$ a uniformly Federer measure on $X$, and let $S$ be as above. For $m\in\mathbb{N}$, let a ball $B=B(x_{0},r_{0})\subset X$ and a continuous map $h:\tilde{B}\to\operatorname{GL}(m,\mathbb{Q}_{S})$ be given, where $\tilde{B}$ stands for $B(x_{0},3^{m}r_{0})$. Now suppose that for some $C,\alpha>0$ and $0<\rho<1$ one has

(i)* for every $\,\Delta\in\mathcal{P}({\mathbb{Z}}_{S},m)$, the function \operatorname{cov}\big{(}h(\cdot)\Delta\big{)} is $(C,\alpha)$ good  on $\tilde{B}$ with respect to $\mu$;*

(ii)* for every $\,\Delta\in\mathcal{P}({\mathbb{Z}}_{S},m)$, \sup_{x\in B\cap\operatorname{supp}\mu}\operatorname{cov}\big{(}h(x)\Delta\big{)}\geq\rho^{rk(\Delta)}.*

Then for any positive $\varepsilon\leq\rho$ one has

[TABLE]

Proof.

The proof goes line by line as Theorem (8.3)[23] using Theorem 5.2 in place of theorem (6.3) of [23]. ∎

6. Applying nondivergence estimates

For any $\mathbf{y}\in\mathbb{Q}_{p}^{n+1}$ we associate a lattice $u_{\mathbf{y}}\mathcal{D}^{n+1}$ in $(\mathbb{Q}_{p}\times\mathbb{R})^{n+1}$, where $u_{\mathbf{y}}$ is defined as

[TABLE]

and $u_{y}^{\infty}=I_{n+1}$. For $t\in\mathbb{N}$ define

[TABLE]

So $g_{t}u_{y}\in\operatorname{GL}_{n+1}(\mathbb{Q}_{p}\times\mathbb{R})$.

We now proceed to connect dynamics with Diophantine properties. For any $v>n$ and $\frac{1}{n+1}>d>c=\frac{v-n}{(n+1)(v+1)}$, define $v_{d}=\frac{n(1+d)+d}{1-(n+1)d}$. Hence $\frac{n(1+c)+c}{1-(n+1)c}=v<v_{d}$. By Proposition 4.1,

[TABLE]

Now define the set

[TABLE]

Then we have

Lemma 6.1**.**

Let $\nu$ be a measure on $\mathbb{Q}_{p}^{n}$. Then the following statements are equivalent:

1. (1)

$\nu(\mathcal{W}^{p}_{v_{d}})=0~{}~{}\forall~{}d>c.$ 2. (2)

$\nu(\widetilde{\mathcal{W}_{v_{d}}})=0~{}~{}\forall~{}d>c.$**

Proof.

We first prove that $(1)$ implies $(2)$. Let $\mathbf{y}\in\widetilde{\mathcal{W}_{v_{d}}},$ then there exists arbitrarily large $t\in\mathbb{N}$ such that

[TABLE]

Observe that

[TABLE]

[TABLE]

and

[TABLE]

So from (6.1) there exist infinitely many $t\in\mathbb{N}$ such that

[TABLE]

which implies that

[TABLE]

for some $\tilde{\mathbf{q}}\in\mathcal{D}^{n+1}$. Hence $\widetilde{\mathcal{W}_{v_{d}}}\subset\mathcal{W}^{p}_{v_{d}}~{}~{}\forall~{}d>c.$

We now prove $(2)$ implies ($1$). We claim that $\mathcal{W}^{p}_{v_{d}}\subset\widetilde{\mathcal{W}_{v_{d}^{\prime}}}$ for some $d>d^{\prime}>c$. Note that

[TABLE]

Finally,

[TABLE]

for some $\tilde{\mathbf{q}}\in\mathcal{D}^{n+1}$. Therefore we have that $\mathcal{W}^{p}_{v_{d}}\subset\widetilde{\mathcal{W}_{v_{d}^{\prime}}}$ and the conclusion follows. ∎

Proposition 6.1**.**

*Take $\mathcal{R}=\mathbb{Q}_{p}\times\mathbb{R}$ and $\mathcal{D}=\mathbb{Z}[1/p]$and $v>n,c:=\frac{v-n}{(n+1)(v+1)}$ as defined in Proposition 4.1. Let $X$ be a Besicovitch metric space and $\mu$ be a uniformly Federer measure on $X$. Denote $\tilde{B}:=B(x,3^{n+1}r)$. Suppose we are given a continuous function $\mathbf{f}:X\mapsto\mathbb{Q}_{p}^{n}$ and $C,\alpha>0$ with the following properties

(i) $x\mapsto\operatorname{cov}(g_{t}u_{\mathbf{f}(x)}\Delta)$is $(C,\alpha)$ good with respect to $\mu\text{ in }\tilde{B}~{}\forall~{}\Delta\in\mathcal{P}(\mathcal{D},n+1),$

(ii) for any $d>c$ there exists $T=T(d)>0$ such that for any integer $t\geq T$ and any $\Delta\in\mathcal{P}(\mathcal{D},n+1)$ one has*

[TABLE]

Then $w_{p}(f_{*}\mu|_{B})\leq v$.

Proof.

We will check that the map $h=g_{t}u_{\mathbf{f}}$ satisfies the assumptions of Theorem 5.3 with respect to the measure $\nu=\mathbf{f}_{*}\mu|_{B}$ where (6.1) is the same as (i) of Theorem 5.3 with $m=n+1$. To check the second condition take $\rho=p^{-\frac{c+d}{2}t}\text{ for }d>c$. Then (6.1) gives (ii) of Theorem 5.3 for all integer $t>T(\frac{c+d}{2}).$ Therefore by Theorem 5.3

[TABLE]

for all but finitely many $t\in\mathbb{N}$.

Using the Borel-Cantelli lemma and (6.3) we have $\nu(\widetilde{\mathcal{W}_{v_{d}}})=0~{}\forall~{}d>c$, which again gives $\nu({\mathcal{W}^{p}_{v_{d}}})=0~{}\forall~{}d>c$ by Lemma 6.1. Since $\lim_{k\to\infty}d_{k}=c$, we have that $\lim_{k\to\infty}v_{d_{k}}=v$ and now using the formula for $v_{d}$, we have that for any $u>v$ there exists $\frac{1}{n+1}>d>c$ such that $u>v_{d_{k}}$. So $\nu(\mathcal{W}^{p}_{u})=0$ because $\mathcal{W}^{p}_{u}\subset\mathcal{W}^{p}_{v_{d_{k}}}$. Hence $w_{p}(\nu)=w_{p}(f_{*}\mu|_{B})\leq v$. ∎

Let us now show that condition $(2)$ in Proposition 6.1 is in fact necessary to conclude Proposition 6.1. This a $p$-adic version of Lemma (4.1) of [20].

Lemma 6.2**.**

Let $\mu$ be a measure on $B$ and take $c,v>0$ as before. Let $\mathbf{f}:B\mapsto\mathbb{Q}_{p}^{n}$ be such that condition (ii) in Proposition 6.1 does not hold. Then

[TABLE]

Proof.

There exists $d>c$ such that condition(2) in Proposition 6.1 does not hold. So there exists a $1\leq j\leq(n+1)$ such that there exists a sequence of natural numbers $t_{i}\to\infty$ and corresponding $\Delta_{i}$ of rank $j$ such that

[TABLE]

Now consider the ball

[TABLE]

Denote $\Delta=g_{t_{i}}u_{\mathbf{f}(x)}\Delta_{i}$ and $D\cap\mathbb{Q}_{S}\Delta=D_{1}$, a ball in $\mathbb{Q}_{S}\Delta$. Then $\lambda_{S}(D_{1})=\mu_{S}(\pi^{\text{-}1}(D_{1}))$ where $\mu_{S}$ is the Haar measure on $\mathbb{Q}_{S}^{j}$ and

[TABLE]

where the $v_{1},v_{2},\cdots,v_{j}$ are taken such that $v_{1}^{\infty},\cdots,v_{j}^{\infty}$ form a orthonormal basis of $(\mathbb{Q}_{s}\Delta)_{\infty}$ and

[TABLE]

Note that $\lambda_{S}$ is the normalized Haar measure on $\mathbb{Q}_{S}\Delta$. Now consider

[TABLE]

Since $v_{1}^{\infty},\cdots,v_{j}^{\infty}$ are orthonormal, we have that $\mu_{\infty}(\tilde{D}_{1}^{\infty})=2^{j}.p^{-jdt_{i}}$.

On the other hand, since $B(0,1)\times B(0,1)\times\cdots B(0,1)\subset\tilde{D}_{1}^{(p)}$,

[TABLE]

So

[TABLE]

and this gives

[TABLE]

Then by the $S$-arithmetic version of Minkowski’s theorem ([23], Lemma 7.7) there is a nonzero vector $v\in g_{t_{i}}u_{\mathbf{f}(x)}\Delta_{i}\cap D_{1}$ i.e. there exists $\tilde{\mathbf{q}}=(q_{0},q_{1},\cdots,q_{n})\in(\mathbb{Z}[1/p])^{n+1}$ such that

[TABLE]

and

[TABLE]

Therefore

[TABLE]

holds for infinitly many $t_{i}\in\mathbb{N}$. Hence by equivalence (4.1)

[TABLE]

Thus there exists $v_{d}>v$ such that

[TABLE]

∎

Throughout the rest of the paper we are going to denote $\mathcal{R}=\mathbb{Q}_{p}\times\mathbb{R}$ and $\mathcal{D}=\mathbb{Z}[1/p]$. One can associate to any nonzero submodule $\Delta\subset\mathcal{D}^{n+1}$ of rank $j$, an element $\mathbf{w}$ of $\bigwedge^{j}(\mathcal{D}^{n+1})$ such that $\operatorname{cov}(\Delta)=c(\mathbf{w})$ and $\operatorname{cov}(g_{t}u_{\mathbf{y}}\Delta)=c(g_{t}u_{\mathbf{y}}\mathbf{w})$. We take $\mathbf{e}_{0},\mathbf{e}_{1},\cdots,\mathbf{e}_{n}\in\mathcal{R}^{n+1}$ as the standard basis where $\mathbf{e}_{i}=(\mathbf{e}_{i}^{p},\mathbf{e}_{i}^{\infty})$ and $\{\mathbf{e}_{i}^{p}\},\{\mathbf{e}_{i}^{\infty}\}$ are the standard basis of $\mathbb{Q}_{p}^{n}$, $\mathbb{R}^{n}$ respectively. We will use the standard basis $\{\mathbf{e}_{I}=\mathbf{e}_{i_{1}}\wedge\cdots\wedge\mathbf{e}_{i_{j}}~{}|I\subset\{0,\cdots,n\}\text{ and }i_{1}<i_{2}<\cdots<i_{j}\}$ of $\bigwedge^{j}\mathcal{R}^{n+1}$. Thus we can write any $\mathbf{w}\in\mathcal{D}^{n+1}$ as $\mathbf{w}=\sum w_{I}\mathbf{e}_{I}$, where $w_{I}\in\mathcal{D}$. Let us note the action of unipotent flows on the coordinates of $\mathbf{w}$.

1. (1)

$u_{\mathbf{y}}^{p}$ leaves $\mathbf{e}_{0}^{p}$ invariant and sends $\mathbf{e}_{i}^{p}$ to $y_{i}\mathbf{e}_{0}^{p}+\mathbf{e}_{i}^{p}$ for $i\geq 1$. 2. (2)

$u_{\mathbf{y}}^{\infty}=I_{n+1}$ leaves everything invariant.

Therefore

[TABLE]

Observe that under the action of $g_{t}^{p}$, $\mathbf{e}_{i}^{p}$s’ are invariant for $i\geq 1$ and $\mathbf{e}_{0}^{p}$ is an eigenvector with eigenvalue $p^{-t}$. Therefore

[TABLE]

On the otherhand $u_{\mathbf{y}}^{\infty}=\operatorname{Id}$ and each $\mathbf{e}_{i}^{\infty}$ is an eigenvector with eigenvalue $p^{-\frac{t}{n+1}}$. Thus

[TABLE]

Therefore for $\mathbf{w}\in\bigwedge^{j}(\mathcal{D}^{n+1}),\mathbf{w}=\sum w_{I}\mathbf{e}_{i}$ where $w_{I}\in\mathcal{D}$ we get

[TABLE]

and

[TABLE]

It will be convenient for us to use the following notations,

[TABLE]

where $\mathbf{c}(\mathbf{w})_{i}=\sum\limits_{\begin{subarray}{c}J\subset\{1,\cdots,n\}\\ \#J=j-1\end{subarray}}w_{J\cup\{i\}}\mathbf{e}^{p}_{J}\in\bigwedge^{j-1}(V_{0})$ and $V_{0}$ is the subspace of $\mathbb{Q}_{p}^{n+1}$ generated by $\mathbf{e}^{p}_{1},\cdots,\mathbf{e}^{p}_{n}$. We may therefore write

[TABLE]

where $y_{0}=1$ and $\pi$ is the orthogonal projection from $\bigwedge^{j}(\mathbb{Q}_{p}^{n+1})\mapsto\bigwedge^{j}(V_{0})$. If $\mathbf{w}$ corresponds to $\Delta\subset\mathcal{D}^{n+1}$, a submodule of rank $j$ then we have that

[TABLE]

Thus,

[TABLE]

where $\tilde{\mathbf{f}}=(1,f_{1},\cdots,f_{n})$. Note that condition (6.2) can be written as

[TABLE]

Now suppose that the $\mathbb{Q}_{p}$-span of the restrictions of $1,f_{1},\cdots,f_{n}$ to $B\cap\operatorname{supp}\mu$ has dimension $s+1$ and choose $g_{1},\cdots,g_{s}:B\cap\operatorname{supp}\mu\mapsto\mathbb{Q}_{p}$ such that $1,g_{1},\cdots,g_{s}$ form a basis of the space. Therefore there exists a matrix $R=(r_{i,j})_{(s+1)\times(n+1)}$ such that $\tilde{\mathbf{f}}(x)=\tilde{\mathbf{g}}(x)R~{}\forall~{}x\in B\cap\operatorname{supp}\mu$ where $\tilde{\mathbf{g}}=(1,g_{1},\cdots,g_{s})$. We can rewrite

[TABLE]

Thus we have that (6.2) is equivalent to

[TABLE]

by equivalence of norms since $1,g_{1},\cdots,g_{s}$ are linearly independent. In order to get rid of the auxiliary variable $t$ we want to apply lemma 4.1. Consider the components of $R\mathbf{c}(\mathbf{w})$:

[TABLE]

So we have that

[TABLE]

Now consider the set

[TABLE]

Lemma 6.3**.**

The set $\mathcal{E}$ as above satisfies the hypotheses of Lemma 4.1.

Proof.

Let $\mathbf{w}_{m}\in\mathcal{S}_{n+1,j}$ be a sequence such that

[TABLE]

for some $M>0$ and

[TABLE]

as $m\to\infty$ where $\mathbf{w}_{m}=\sum w^{(m)}_{I}e_{I}$. For $J\subset\{1,\cdots,n\},$ such that $\#J=j-1$, denote $\tilde{\mathbf{w}}_{J}^{(m)}:=(w^{(m)}_{J\cup\{0\}},\cdots,w^{(m)}_{J\cup\{n\}})$. We have that for every $m,J$ and $k=1,\cdots,n,$

[TABLE]

Moreover for $\varepsilon>0~{}\exists~{}N_{\varepsilon}\in\mathbb{N}$ such that $\forall~{}m\geq N_{\varepsilon}$,

[TABLE]

Since $\tilde{\mathbf{f}}(x)=\tilde{\mathbf{g}}(x)R~{}~{}\forall~{}x\in B\cap\operatorname{supp}\mu$ we have $r_{00}=1\text{ and }r_{i0}=0$ otherwise. This implies that

[TABLE]

for some $M_{1}>0$ and for every $J$. Therefore, for $\tilde{\mathbf{w}}_{J}^{(m)}$,

[TABLE]

for some $M_{1}>0$ and

[TABLE]

Now applying Lemma 3.1 we can conclude that first hypothesis is satisfied. The second hypothesis is satisfied since $(0,\|R\mathbf{c}(\mathbf{w})\|_{p}\|\mathbf{w}\|_{\infty})\in\mathcal{E}$ implies that for $u\in\mathbb{N}\text{ with }p\nmid u,(0,u\|R\mathbf{c}(\mathbf{w})\|_{p}\|\mathbf{w}\|_{\infty})\in\mathcal{E}$. ∎

Here, in terms of Lemma 4.1, $a=\frac{n+1-j}{n+1},b=\frac{j}{n+1}$. Since the set $\mathcal{E}$ satisfies the hypotheses of Lemma 4.1, we conclude that the condition (6.14) is equivalent to

[TABLE]

Moreover $\lim_{k\to\infty}d_{k}=c$ implies that $\lim_{k\to\infty}u_{d_{k}}=\frac{v-j+1}{j}$. Therefore condition (6.15) is equivalent to

[TABLE]

The proof of the Theorem 6.2 now goes as in [20], we repeat it for the sake of completeness. Note that we have that $R$ is a matrix depending on the ball $B$, the measure $\mu$ and the map $\mathbf{f}$ such that

[TABLE]

Suppose

[TABLE]

Let $\dim\mathcal{L}=s$ and let

[TABLE]

where as usual we have that $\tilde{\mathbf{h}}:=(1,h_{1},\cdots,h_{n})$ and $\tilde{\mathbf{x}}:=(1,x_{1},\cdots,x_{s})$. Then $\mathbf{g}=\mathbf{h}^{\text{-}1}\circ\mathbf{f}$ generates the space of $\mathbb{Q}_{p}$ span of the restrictions of $1,f_{1},\cdots,f_{n}$ to $B\cap\operatorname{supp}\mu$ and satisfies $\tilde{\mathbf{f}}(x)=\tilde{\mathbf{g}}(x)R~{}\forall~{}x\in B\cap\operatorname{supp}\mu$. Therefore condition (6.16)$\iff$ (6.14) $\iff$condition (6.2) becomes a property of the subspace and in particular $R$ can be chosen uniformly for all measures $\mu$, ball B and measure $\mu$ and $\mathbf{f}$ the function as long as (6.17) holds.

Theorem 6.1**.**

*Let $\mu$ be a Federer measure on a Besicovitch metric space $X,\mathcal{L}$ an affine subspace of $\mathbb{Q}_{p}^{n}$, and let $\mathbf{f}:X\to\mathcal{L}$ be a continuous map such that $(\mathbf{f},\mu)$ is good and nonplanar i.e (6.17) holds for all open balls B with $\mu(B)>0$. Then the following are equivalent for $v\geq n$ and $c_{v}=\frac{v-n}{(n+1)(v+1)},$

*(i)

[TABLE]

(ii)

[TABLE]

(iii)

[TABLE]

(iv)

[TABLE]

Proof.

We have already observed that (6.22) holds if and only if so does (6.21), and that (6.20) implies (6.19) by definition. It remains to show that (6.21) $\implies$ (6.20) and that (6.19) $\implies$ (6.22). Assume (6.21) and since $\mu$ is Federer and $(\mathbf{f},\mu)$ is good, we have for $\mu$-a.e $x\in X$ has a neighbourhood $V$ such that $(\mathbf{f},\mu)$ good and $\mu$ is $D$-federer on $V$. Choose a ball $B=B(x,r)$ of positive measure such that the dilated ball $\tilde{B}=B(x,3^{n+1}r)$ is contained in $V$. Since we have already noticed in (6.14) that $\operatorname{cov}(g_{t}u_{\mathbf{f}(x)}\Delta)$ is $\max$ of norm of linear combinations of $1,f_{1},\cdots,f_{n}$ condition (i)of (6.1) is satisfied. And (6.21) is equivalent to second hypothesis (6.2) in (6.1). Therefore we can conclude (6.20).

Suppose (6.22) does not hold then equivalently condition (6.2) does not hold and by Lemma 6.2 it follows that

[TABLE]

for some $u>v$. ∎

The upshot of the last theorem is that (6.21) and (6.22) do not involve any of $\mathbf{f},\mu,X$. So if conditions (6.19), (6.20) hold for some $\mathbf{f},\mu,X$ satisfying the hypotheses of the Theorem then they hold any other $\mathbf{f}^{\prime},\mu^{\prime},X^{\prime}$ satisfying the hypotheses of this theorem. So for any two $\mathbf{f},\mu,X$ and $\mathbf{f}^{\prime},\mu^{\prime},X^{\prime}$ satisfying the hypotheses of this theorem, we have that $w_{p}(\mathbf{f}_{*}\mu)=w_{p}(\mathbf{f}^{\prime}_{*}\mu^{\prime})$.

Theorem 6.2**.**

Let $\mu$ be a Federer measure on a Besicovitch metric space $X,\mathcal{L}$ an affine subspace of $\mathbb{Q}_{p}^{n}$, and let $\mathbf{f}:X\to\mathcal{L}$ be continuous map such that $(\mathbf{f},\mu)$ is good and nonplanar in $\mathcal{L}$. Then

[TABLE]

Proof.

Let us take $\nu=\lambda$ (Haar measure) on $\mathbb{Q}_{p}^{s}$ and $\mathbf{h}$ as in (6.18). Then by definition, $w(\mathcal{L})=w_{p}(\mathbf{h}_{*}\nu)$. Since $\mathbf{h},\nu$ and $\mathbb{Q}_{p}^{s}$ satisfy the hypotheses of Theorem 6.2, we have $w_{p}(\mathbf{f}_{*}\mu)=w_{p}(\mathcal{L})$ by the previous discussion. And we already have from definitions that

[TABLE]

Therefore to conclude the theorem it is enough to show that

[TABLE]

But this follows by taking $v=\inf\{w_{p}(\mathbf{y})~{}|~{}\mathbf{y}\in\mathcal{L}\}$ and $\mathbf{h}$ in Theorem 6.2; condition (6.19) automatically holds. ∎

Corollary 6.1**.**

Let $\mathcal{L}$ be an $s$-dimensional affine subspace of $\mathbb{Q}_{p}^{n}$. Then

[TABLE]

In view of Proposition $3.1$, we have

Corollary 6.2**.**

Let $\mu$ be a Federer measure on a Besicovitch metric space $X,\mathcal{L}$ an affine subspace of $\mathbb{Q}_{p}^{n}$, and let $\mathbf{f}:X\to\mathcal{L}$ be continuous map such that $(\mathbf{f},\mu)$ is good and nonplanar in $\mathcal{L}$. Then

[TABLE]

7. Computation of Higher Diophantine Exponents

In this section we want to relate condition (6.16) in terms of Diophantine conditions of a parametrizing matrix of the $s$ dimensional subspace $\mathcal{L}$. Set

[TABLE]

where $A=(a_{ij})_{\begin{subarray}{c}i=0,\cdots s\\ j=s+1,\cdots,n\end{subarray}}=\begin{bmatrix}\mathbf{a}_{0}\\ \vdots\\ \mathbf{a}_{s}\end{bmatrix}\in M_{s+1,n-s}$. Then we can write

[TABLE]

Let $\pi_{\bullet}$ be the projection from $\bigwedge^{j}\mathbb{Q}_{p}^{n+1}\bigoplus\bigwedge^{j}\mathbb{R}^{n+1}$ to

[TABLE]

where $F\langle\mathbf{e}_{s+1},\cdots,\mathbf{e}_{n}\rangle$ denotes the $F$ span of $\mathbf{e}_{s+1},\cdots,\mathbf{e}_{n}$, the last $n-s$ vectors of standard basis of $F^{n+1}$ for a field $F$.

Lemma 7.1**.**

Suppose that $\|R_{A}\mathbf{c}(\mathbf{w})\|_{p}\|\mathbf{w}\|_{\infty}\leq 1$ for some $\mathbf{w}\in\mathcal{S}_{n+1,j}$. Then $\|\mathbf{w}\|_{p}\|\mathbf{w}\|_{\infty}\ll 1+\|\pi_{\bullet}(\mathbf{w})\|_{p}\|\mathbf{w}\|_{\infty}$.

Proof.

Suppose the smallest index of $I\subset\{1,\cdots,n\}(\#I=j)$ is $m$. If $m>s$ then we have that

[TABLE]

On the other hand, when $m\leq s$ we have

[TABLE]

Now the same argument can be applied to the each of the components $|\langle\mathbf{e}^{p}_{J},\mathbf{w}\rangle|_{p}\|\mathbf{w}\|_{\infty}$ where the smallest index is at least $m+1$. After at most $s+1$ steps the process terminates and this concludes the proof. ∎

Note that for $\mathbf{w}\in\mathcal{S}_{n+1,j}$ with $j>n-s$ we have that $\pi_{\bullet}(\mathbf{w})=0$. Since

[TABLE]

we have that

[TABLE]

for $u>0$ and for all arbitrarily large $\|\pi(\mathbf{w})\|_{p}\|\mathbf{w}\|_{\infty}$. Thus condition (6.16) holds for $j>n-s$. Therefore while computing the exponent $w_{p}(\mathcal{L}),$ according to Corollary 6.1, the subgroups of rank greater than $n-s$ are irrelevant. We now define higher Diophantine exponents.

Definition 7.1**.**

For each $j=1,\cdots,n-s,$ let

[TABLE]

We now observe

Corollary 7.1**.**

For an s-dimensional subspace $\mathcal{L}$ parametrized by $A$ as in (7.1)

[TABLE]

Proof.

By Lemma 7.1 we can conclude that for some $v$ condition (6.16) holds if and only if $\max_{j=0,\cdots,n-s}w^{p}_{j}(A)\leq v$. And thus by (6.1), we can conclude this corollary. ∎

Lemma 7.2**.**

$w^{p}_{1}(A)=w_{p}(A).$**

Proof.

Take $\mathbf{w}=\begin{bmatrix}q_{0},\\ q_{1}\\ \vdots\\ q_{s}\\ q_{s+1}\\ \vdots\\ q_{n}\end{bmatrix}=\tilde{\mathbf{q}}\in\mathcal{D}^{n+1}\setminus\{0\}=\mathcal{S}_{n+1,1}$ and denote $\mathbf{q}_{0}=\begin{bmatrix}q_{0}\\ \vdots\\ q_{s}\end{bmatrix}$ and $\mathbf{q}=\begin{bmatrix}q_{s+1}\\ \vdots\\ q_{n}\end{bmatrix}$. Then

[TABLE]

Hence the definition of both exponents coincide. ∎

Theorem 7.1**.**

Suppose $\mathcal{L}$ is an $s=n-1$ dimensional subspace parametrized by $R_{A}$ as in (7.1). Then

[TABLE]

Proof.

The Theorem follows directly from Lemma 7.2 and Corollary 7.1. ∎

Lemma 7.3**.**

Let $\mathcal{L}$ be parametrized by $R_{A}$ as in (7.1). Then for any $v<w_{p}(A)$ there exists arbitrary large $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$ such that

[TABLE]

Proof.

Note that having arbitrarily large $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$ is not the same as infinitely many $\tilde{\mathbf{q}}\in\mathcal{D}^{n+1}$. Since $v<w_{p}(A)$ there exists $\varepsilon>0$ such that

[TABLE]

Hence there exists arbitrary large $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$ such that

[TABLE]

and $\tilde{\mathbf{q}}=(\mathbf{q}_{0},\mathbf{q})=\left(\begin{bmatrix}q_{0}\\ \vdots\\ q_{s}\end{bmatrix},\begin{bmatrix}q_{s+1}\\ \vdots\\ q_{n}\end{bmatrix}\right)\in\mathcal{D}^{s+1}\times\mathcal{D}^{n-s}$.

For any $\mathbf{y}=(\mathbf{x},\tilde{\mathbf{x}}A)\in\mathcal{L}$ where $\tilde{\mathbf{x}}=(1,\mathbf{x})\in\mathbb{Q}_{p}^{s+1}$ we can write

[TABLE]

for arbitrary large $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$. This implies that we also have

[TABLE]

for arbitrary large $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$ where $\mathbf{q}_{1}=\begin{bmatrix}q_{1}\\ \vdots\\ q_{n}\end{bmatrix}$.

We also have

[TABLE]

which implies that

[TABLE]

Then we have that

[TABLE]

for arbitrary large $\|\mathbf{q}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$, hence for arbitrary large $\|\mathbf{q}_{1}\|_{p}\|\tilde{\mathbf{q}}\|_{\infty}$, thereby concluding the proof. ∎

An immediate consequence of above lemma is the following corollary.

Corollary 7.2**.**

Let $\mathcal{L}$ is parametrized by $R_{A}$ as in (7.1). Then

[TABLE]

Another observation is when in the parametrizing matrix $A$ with more than one column and all columns are rational multiple of one column then $w_{p}(A)=\infty$ and so is $w_{p}(\mathcal{L})$. Combining this observation with Theorem 7.1 we can conclude the following:

Theorem 7.2**.**

Let $\mathcal{L}$ be an affine subspace parametrized by $R_{A}$ as in (7.1). If all the columns are rational multiples of one column then

[TABLE]

The calculation of symmetries of higher order exponents is exactly the same as in case of $\mathbb{R}$ as discussed in [20]. So we will just state them in the $p$-adic setup.

Lemma 7.4**.**

For any $A\in M_{s+1,n-s}$ and all $\mathbf{w}\in\mathcal{S}_{n+1,{j}}$, $2\leq j\leq n-s$, one has

[TABLE]

and

[TABLE]

The Lemma enables us to conclude that $w_{j}^{p}(A)$ is symmetric under any row operation. The next lemma allows to consider other row operations namely the multiplication by nonzero rationals and adding one row to another and transposition of rows.

Lemma 7.5**.**

Let $A^{\prime}=BA$ for some $B\in\operatorname{GL}_{s+1}(\mathbb{Q})$; in other words, $A^{\prime}$ can be obtained from $A$ by a sequence of elementary row operations with rational coefficients. Then $w^{p}_{j}(A^{\prime})=w^{p}_{j}(A)$ for all $j$.

Now we will wrap up this section by stating the last lemma and a theorem similar to Theorem 7.2. for rational multiples of rows.

Lemma 7.6**.**

Suppose that $A$ has more than one row, and let $A^{\prime}$ be the matrix obtained from $A$ by removing one of its rows. Then $w_{j}^{p}(A^{\prime})\geq w_{j}^{p}(A)$ for all $j$. If in addition the removed row is a rational linear combination of the remaining rows, then $w_{j}^{p}(A^{\prime})=w_{j}^{p}(A)$ for all $j$.

As a consequence we have the following theorem.

Theorem 7.3**.**

Let $\mathcal{L}$ be an affine subspace parametrized by $R_{A}$ as in (7.1). If all the rows are rational multiples of one row then one has

[TABLE]

The lemmata and Theorem stated above are proved by a verbatim repetition of the arguments of Kleinbock [20], and so we omit the proofs.