# Counting rational points close to $p$-adic integers and applications in
Diophantine approximation

Benjamin Ward  Department of Mathematics, University of York, Heslington,
York, YO10 5DD, United Kingdom<EMAIL_ADDRESS>
(Date: January, 2020)

###### Abstract.

We find upper and lower bounds on the number of rational points that are
$\psi$-approximations of some $n$-dimensional $p$-adic integer. Lattice point
counting techniques are used to find the upper bound result, and a Pigeon-hole
principle style argument is used to find the lower bound result. We use these
results to find the Hausdorff dimension for the set of $p$-adic weighted
simultaneously approximable points intersected with $p$-adic coordinate
hyperplanes. For the lower bound result we show that the set of rational
points that $\tau$-approximate a $p$-adic integer form a set of resonant
points that can be used to construct a local ubiquitous system of rectangles.

##  1\. Introduction

The study of rational points on algebraic varieties, usually called
Diophantine geometry, has a wide variety of applications in many areas of
mathematics. A variation of this is the study of rational points that lie
close to such algebraic varieties. In the setting of $\mathbb{R}^{n}$ there
has been many results of this type, including counts on the number of rational
points close to curves [7, 40, 37, 38, 27] and manifolds [4, 13, 25, 26]. In
the $p$-adic setting less is known. In [2, 3] a bound on the number of
rational points that lie on the curve
$\mathcal{C}_{f}=\\{(x,x^{2},\dots,x^{n}):x\in\mathbb{Z}_{p}\\}$ were found,
but as yet no other results are available. In this paper we provide an upper
and lower bound on the number of rational points within a small neighbourhood
of a $p$-adic integer. Such result allows us to find bounds on the number of
rational points close to $p$-adic coordinate hyperplanes.

Fix a prime number $p\in\mathbb{N}$ and let $|.|_{p}$ denote the $p$-adic
norm. Define the set of $p$-adic numbers $\mathbb{Q}_{p}$ as the completion of
$\mathbb{Q}$ with respect to the $p$-adic norm. Denote by
$\mathbb{Z}_{p}:=\\{x\in\mathbb{Q}_{p}:|x|_{p}\leq 1\\}$ the ring of $p$-adic
integers. Let $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$, $N\in\mathbb{N}$, and
$\Psi=(\psi_{1},\dots,\psi_{n})$ be an $n$-tuple of approximation functions of
the form $\psi_{i}:\mathbb{N}\to\mathbb{R}_{+}$, with $\psi_{i}(q)\to 0$ as
$q\to\infty$ for each $1\leq i\leq n$. We provide bounds on the cardinality of
the set

$\mathcal{Q}(\boldsymbol{x},\Psi,N):=\left\\{(q_{0},q_{1},\dots,q_{n})\in\mathbb{Z}^{n+1}:\begin{array}[]{c}0<q_{0}\leq
N,\\\ \max_{1\leq i\leq n}|q_{i}|\leq N,\\\
\end{array}\,\left|q_{0}x_{i}-q_{i}\right|_{p}<\psi_{i}(N),\,1\leq i\leq
n\right\\}.$

If the approximation functions $\psi_{i}$ are of the form
$\psi_{i}(q)=q^{-\tau_{i}}$ for some vector
$\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{n})\in\mathbb{R}^{n}_{>0}$ we will
use the notation $\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau},N)$. Note that
to get a result for general $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$ we must
apply some conditions. For example, if $\boldsymbol{x}\in\mathbb{Q}^{n}$ then
for sufficiently large $N\in\mathbb{N}$ we have that
$\\#\mathcal{Q}(\boldsymbol{x},\Psi,N)\asymp N^{2}$ for any $\Psi$. Here
$a\asymp b$ means there exists constants $c_{1},c_{2}\in\mathbb{R}_{>0}$ such
that $c_{1}b\leq a\leq c_{2}b$. Conversely, if $\boldsymbol{x}$ is badly
approximable each approximation function satisfies
$\psi_{i}(q)<q^{-1-\frac{1}{n}-\epsilon}$ for some $\epsilon>0$, then
$\\#\mathcal{Q}(\boldsymbol{x},\Psi,N)\ll 1$. In order to obtain good bounds
on the cardinality of $\mathcal{Q}(\boldsymbol{x},\Psi,N)$ we use the
Diophantine exponent $\tau(\boldsymbol{x})$ defined as

$\tau(\boldsymbol{x}):=\sup\left\\{\sum_{i=1}^{n}\tau_{i}:|q_{0}x_{i}-q_{i}|_{p}<Q^{-\tau_{i}},\,\,\text{
for i.m. }\,Q\in\mathbb{N}\text{ with }|q_{i}|\leq Q\right\\}.$

By a Theorem of Mahler [39] we have that for all $x\in\mathbb{Z}_{p}$,
$\tau(x)\geq 2$. Further, by a result of Jarnik [30] we have that
$\tau(\boldsymbol{x})=n+1$ for almost all
$\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$, with respect to the $n$-dimensional
Haar measure $\mu_{p,n}$ on $\mathbb{Q}_{p}$, normalised by
$\mu_{p,n}(\mathbb{Z}_{p}^{n})=1$.

We have the following result on the cardinality of $\mathcal{Q}(x,\psi,N)$ for
general $x\in\mathbb{Z}_{p}$.

######  Lemma 1.1.

Let $x\in\mathbb{Z}_{p}$ with Diophantine exponent $\tau(x)$ and let
$\psi(q)=q^{-\tau}$ for some $\tau\in\mathbb{R}_{+}$ with
$\max\\{1,\tau(x)-1\\}<\tau<\tau(x)$. Then for any $\epsilon>0$ there exists
sufficiently large $N_{0}\in\mathbb{N}$ such that for all $N\geq N_{0}$

$\\#\mathcal{Q}(x,\tau,N)\leq N^{\tau(x)-\tau+\epsilon}.$

Note by our previous remark on the Diophantine exponent that for almost all
$x\in\mathbb{Z}_{p}$ we have $\tau(x)=2$, so the above lemma reads that for
$\psi(q)=q^{-\tau}$ with $1<\tau<2$, then for almost all $x\in\mathbb{Z}_{p}$

$\\#\mathcal{Q}(x,\psi,N)\leq N^{2-\tau+\epsilon}.$

While Lemma 1.1 gives us an upper bound for all $x\in\mathbb{Z}_{p}$, provided
the approximation function $\psi$ is ’close’ to the function related to the
Diophantine exponent, the bound given has an extra $Q^{\epsilon}$ term, which
we believe is unnecessary. The following theorem offers an improvement in this
respect.

######  Theorem 1.2.

Let $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$ and suppose that
$\tau(\boldsymbol{x})=n+1$. Let $\Psi$ be an $n$-tuple of approximation
functions with each

$q^{-1-\frac{1}{n}+\epsilon}<\psi_{i}(q)<q^{-1},\quad 1\leq i\leq n,$

for some $\epsilon>0$. Then there exists $N_{0}\in\mathbb{N}$ such that for
all $N\geq N_{0}$,

$\\#\mathcal{Q}(\boldsymbol{x},\Psi,N)\leq
C_{1}N^{n+1}\prod_{i=1}^{n}\psi_{i}(N),$

where

$C_{1}=\max\left\\{3(6\sqrt{n})^{n},\frac{(n+2)!\pi^{n/2}\sqrt{n}^{n+1}}{\Gamma\left(\frac{n}{2}+1\right)}\right\\}.$

Again, as with Lemma 1.1, we can deduce that the above upper bound is true for
almost all $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$. This type of result has
already been proven in the real case (see Lemma 6.1 of [8]). In the case where
the approximation functions are of the form $\psi_{i}(q)=q^{-\tau_{i}}$ then
the theorem reads: if

$\sum_{i=1}^{n}\tau_{i}<n+1,\quad\text{ and }\quad\tau_{i}>1,$

then for any $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$ with
$\tau(\boldsymbol{x})=n+1$,

$\\#\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau},N)\leq
C_{1}N^{n+1-\sum_{i=1}^{n}\tau_{i}}.$

Lastly, we have the following lemma which provides a complimentary lower bound
to the previous two results.

######  Lemma 1.3.

Let $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$ and

$\sum_{i=1}^{n}\tau_{i}<n+1,\quad\text{and}\quad\tau_{i}>1$

for each $1\leq i\leq n$. Then there exists $N_{0}\in\mathbb{N}$ such that for
all $N\geq N_{0}$ we have that

$\\#\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau},N)\geq\frac{1}{p}N^{n+1-\sum_{i=1}^{n}\tau_{i}}-1.$

As with Theorem 1.2, the equivalent version of this result in $\mathbb{R}^{n}$
has previously been proven, (see Lemma 3 of [36]). Further, as
$\sum_{i=1}^{n}\tau_{i}<n+1$ we can choose $N$ large enough such that

$\\#\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau},N)\geq\frac{1}{2p}N^{n+1-\sum_{i=1}^{n}\tau_{i}}.$

Thus combining this with Theorem 1.2 we have the expected result that
$\\#\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau},N)\asymp
N^{n+1-\sum_{i=1}^{n}\tau_{i}}$.

The proofs of Lemma 1.1 and Lemma 1.3 use elementary techniques. The proof of
Theorem 1.2 is more substantial and uses $p$-adic approximation lattices and
lattice counting techniques. Prior to the proofs of these results we give an
example of their applications in Diophantine approximation.

##  2\. $p$-adic Diophantine approximation

As an application of the main results in the previous section we consider the
set of $p$-adic simultaneously approximable points over coordinate
hyperplanes. Define the set of weighted simultaneously approximable points as
follows. For an $n$-tuple of approximation functions
$\Psi=(\psi_{1},\dots,\psi_{n})$ and $q_{0}\in\mathbb{N}$ let

$\mathfrak{A}_{q_{0}}(\Psi)=\underset{1\leq i\leq n}{\bigcup_{|q_{i}|\leq
q_{0},\,\,gcd(q_{i},q_{0})=1}}\left\\{\boldsymbol{x}\in\mathbb{Z}_{p}^{n}:\left|x_{i}-\frac{q_{i}}{q_{0}}\right|_{p}<\psi_{i}(q_{0})\right\\},$

where $\boldsymbol{x}=(x_{1},\dots,x_{n})$. Define the set of weighted
$\Psi$-approximable $p$-adic points as

$\mathfrak{W}_{n}(\Psi):=\limsup_{q_{0}\to\infty}\mathfrak{A}_{q_{0}}(\Psi).$

If the approximation functions have the form $\Psi=(\psi,\dots,\psi)$ then we
will use the notation $\mathfrak{W}_{n}(\Psi)=\mathfrak{W}_{n}(\psi)$, and if
$\psi_{i}(q)=q^{-\tau_{i}}$ for each $1\leq i\leq n$ and for some exponents of
approximation
$\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{n})\in\mathbb{R}^{n}_{>0}$ we will
use the notation $\mathfrak{W}_{n}(\Psi)=\mathfrak{W}_{n}(\boldsymbol{\tau})$.
By considering the Dirichlet style theorem for the set
$\mathfrak{W}_{n}(\boldsymbol{\tau})$ we have that
$\mathfrak{W}_{n}(\boldsymbol{\tau})=\mathbb{Z}_{p}^{n}$ provided that
$\sum_{i=1}^{n}\tau_{i}\leq n+1$. There are also a variety of results on the
Haar measure of $p$-adic approximable points. The classical result was
originally proven by Jarnik [30], and since then there has been results in the
linear forms [32], weighted [11], and Duffin-Schaeffer [22] cases.

For sets of zero Haar measure we use Hausdorff measure and Hausdorff dimension
to provide more accurate notions of size. We briefly recap the definition and
notation of Hausdorff measure and dimension. For a metric space $(X,d)$, a set
$U\subset X$, and $\rho>0$, define a $\rho$-cover of $U$ as a sequence of
balls $\\{B_{i}\\}$ such that $U\subset\bigcup_{i}B_{i}$ and for all balls
$r(B_{i})\leq\rho$, where $r(.)$ denotes the radius of the ball. Define a
dimension function $f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ as an increasing
continuous function with $f(r)\to 0$ as $r\to 0$. Define the $f$-Hausdorff
measure as

$\mathcal{H}^{f}(U)=\lim_{\rho\to
0^{+}}\inf\left\\{\sum_{i}f(r(B_{i})):\\{B_{i}\\}\,\,\text{ is a $\rho$-cover
of}\,U\right\\},$

where the infimum is taken over all $\rho$-covers of $U$. When the dimension
function $f(x)=x^{s}$ we will use the notation
$\mathcal{H}^{f}=\mathcal{H}^{s}$. Define the Hausdorff dimension as

$\dim U=\inf\\{s\geq 0:\mathcal{H}^{s}(U)=0\\}.$

In [11] it was proven that, for $\sum_{i=1}^{n}\tau_{i}>n+1$,

$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})=\min_{1\leq i\leq
n}\left\\{\frac{n+1+\sum_{j=i}^{n}(\tau_{i}-\tau_{j})}{\tau_{i}}\right\\}.$

It would be desirable to obtain equivalent measure results for simultaneous
$p$-adic approximable points over manifolds. In [31] Kleinbock and Tomanov
proved the extremality of $p$-adic manifolds provided some non-degeneracy
conditions are satisfied. Generally a manifold
$\mathcal{M}\subset\mathbb{Z}_{p}^{n}$ is said to be extremal if for almost
all points $\boldsymbol{x}\in\mathcal{M}$, with respect to the induced Haar
measure of the manifold, we have that $\tau(\boldsymbol{x})=n+1$ (see [31] for
more details). There are a variety of results for $p$-adic dual approximation,
see for example [5, 9, 16, 19, 33], however results in the simultaneous case
are lacking. Recently Oliveira [34] produced a Khintchine-style Theorem for
simultaneous $p$-adic approximation with denominators coming from $p$-adic
balls. This result has a similar style to our result, with the difference
being that our denominators come from a ball with radius tending to zero,
rather than a fixed constant. Other than this there are relatively few
Khintchine-style results.

For the Hausdorff dimension there are recent results on simultaneously
approximable points over the Veronese curve,
$\mathcal{C}_{f}=\\{(x,x^{2},\dots,x^{n}):x\in\mathbb{Z}_{p}\\}$, provided the
approximation functions have sufficiently large Diophantine exponents [3, 17].
In [11] a lower bound for the Hausdorff dimension was found for general
$n$-dimensional normal curves. A key reason the upper bound could not be
obtained was a lack in results on the behaviour of rational points close to
$p$-adic curves. The main results of this paper provide us with a good
understanding of the behaviour of rational points close to coordinate
hyperplanes. The results of this section are closely related to a variety of
results in the real case on Diophantine approximation over coordinate
hyperplanes, see [10, 35, 36].

For a $p$-adic integer $\boldsymbol{\alpha}\in\mathbb{Z}_{p}^{m}$ for $1\leq
m\leq n-1$ define the coordinate hyperplane

$\Pi_{\boldsymbol{\alpha}}:=\\{(x_{1},\dots,x_{d},\boldsymbol{\alpha}):(x_{1},\dots,x_{d})\in\mathbb{Z}_{p}^{d}\\}\subset\mathbb{Z}_{p}^{n},$

where $n=d+m$. For the set
$\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}$ we have the
trivial result that

$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}\leq\dim\Pi_{\boldsymbol{\alpha}}=n-m,$

with equality when $\sum_{i=1}^{n}\tau_{i}\leq n+1$. In this paper we prove
the following result on the Hausdorff dimension of
$\mathcal{W}_{n}(\Psi)\cap\Pi_{\boldsymbol{\alpha}}$.

######  Theorem 2.1.

Let $\Pi_{\boldsymbol{\alpha}}$ be a coordinate hyperplane of
$\mathbb{Z}_{p}^{n}$, let $\boldsymbol{\alpha}\in\mathbb{Z}_{p}^{m}$ satisfy
$\tau(\boldsymbol{\alpha})=m+1$. Let
$\boldsymbol{\tau}=(\tau_{1},\dots,\tau_{n})\in\mathbb{R}^{n}_{+}$ be a weight
vector with the properties that

$\sum_{i=1}^{m}\tau_{d+i}<m+1,\quad\sum_{i=1}^{n}\tau_{i}>n+1,\quad\tau_{i}>1,$

for all $1\leq i\leq n$. Then

$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}=\min_{1\leq
i\leq d}\left\\{\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}+\underset{1\leq j\leq
d}{\sum_{\tau_{j}\leq\tau_{i}}}(\tau_{i}-\tau_{j})}{\tau_{i}}\right\\}=s.$

Further

$\mathcal{H}^{s}\left(\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}\right)=\infty.$

######  Remark 2.2.

The constraints on $(\tau_{d+1},\dots,\tau_{n})$ ensure that we can apply
Theorem 1.2. The condition that $\sum_{i=1}^{n}\tau_{i}>n+1$ ensures that we
do not include the trivial case when
$\mathfrak{W}_{n}(\boldsymbol{\tau})=\mathbb{Z}_{p}^{n}$, in which case
$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}=n-m$.

######  Remark 2.3.

In the special case where the approximation functions are the same i.e.
($\boldsymbol{\tau}=(\tau,\dots,\tau)$), then we have that, for
$1+\frac{1}{n}<\tau<1+\frac{1}{m}$,

$\dim\mathcal{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}=\frac{n+1}{\tau}-m.$

This gives us the expected dimension of the set of approximable points
$\mathcal{W}_{n}(\boldsymbol{\tau})$ less the codimension of the hyperplane
$\Pi_{\boldsymbol{\alpha}}$.

######  Remark 2.4.

We can use the same style of proof used to prove the upper bound of Theorem
2.1, in combination with Lemma 1.1 rather than Theorem 1.2, to prove that for
any $\alpha\in\mathbb{Z}_{p}$ and approximation exponent
$\max\\{1,\tau(\alpha)-1\\}<\tau_{n}<\tau(\alpha)$ we have that

$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\alpha}\leq\min_{1\leq i\leq
n-1}\left\\{\frac{n+\tau(\alpha)-1-\tau_{n}+\underset{j\neq
n}{\sum_{\tau_{j}\leq\tau_{i}}}(\tau_{i}-\tau_{j}),}{\tau_{i}}\right\\}.$

Proving the corresponding lower bound of this result is currently beyond our
reach.

For general approximation functions $\Psi=(\psi_{1},\dots,\psi_{n})$, let

(1) $\psi_{i}^{*}=\lim_{q\to\infty}\frac{-\log(\psi(q))}{\log q}.$

Providing the limits exists and are positive finite for each $1\leq i\leq n$
then define $\Psi^{*}=(\psi^{*}_{1},\dots,\psi_{n}^{*})$.

######  Corollary 2.5.

Let $\Psi=(\psi_{1},\dots,\psi_{n})$ be an $n$-tuple of approximation
functions with each $\psi_{i}$ having positive finite limit (1). If $\Psi^{*}$
satisfy the same conditions as in Theorem 2.1, then for all
$\boldsymbol{\alpha}\in\mathbb{Z}_{p}^{m}$ with
$\tau(\boldsymbol{\alpha})=m+1$,

$\dim\mathfrak{W}_{n}(\Psi)\cap\Pi_{\boldsymbol{\alpha}}={\min_{1\leq i\leq
d}}\left\\{\frac{n+1-\sum_{i=1}^{m}\psi^{*}_{d+i}+\underset{1\leq j\leq
d}{\sum_{\psi^{*}_{j}\leq\psi^{*}_{i}}}(\psi^{*}_{i}-\psi^{*}_{j})}{\psi^{*}_{i}}\right\\}.$

The corollary easily follows from the observation that by the definition of
(1) there exists sufficiently large $q\in\mathbb{N}$ such that

$q^{-\psi^{*}_{i}-\epsilon_{i}}\leq\psi_{i}(q)\leq
q^{-\psi^{*}_{i}+\epsilon_{i}}$

for all $1\leq i\leq n$ and $\epsilon=(\epsilon_{1},\dots,\epsilon_{n})>0$
with $\epsilon_{i}\to 0$ as $q\to\infty$. And so

$\mathfrak{W}_{n}(\Psi^{*}+\epsilon)\subseteq\mathfrak{W}_{n}(\Psi)\subseteq\mathfrak{W}_{n}(\Psi^{*}-\epsilon).$

Letting $\epsilon\to 0$ we obtain the desired result. Note that while
Corollary 2.5 provides a result for general $\Psi$ with components satisfying
(1), there are many functions where such limits do not exist.

The following section provides auxiliary results needed to prove Theorem 2.1.
In particular the framework for the Mass Transference Principle from
rectangles to rectangles [41] is provided. This result is crucial in finding
the lower bound result of Theorem 2.1.

##  3\. Auxiliary results

We provide a brief set of known results that we will use in the proof of
Theorem 2.1. The first result we state can be considered as the $p$-adic
version of Minkowski’s theorem for systems of linear forms. The proof is a
straightforward application of the pigeon-hole principle and can be found in
[11].

######  Lemma 3.1.

Let $L_{i}(\boldsymbol{x}):\mathbb{Z}_{p}^{n}\to\mathbb{Z}_{p}$, with
$i=1,\dots,n$, be linear forms with $p$-adic integer coefficients. Let
$\sum_{i=1}^{n}\tau_{i}=n+1$ for $\tau_{i}\in\mathbb{R}_{+}$, and $H\geq 1$.
Then there exists a non-zero rational integer vector
$\boldsymbol{x}=(x_{0},x_{1},\dots,x_{n})$ with

$\max_{0\leq i\leq n}|x_{i}|\leq H$

satisfying the system of inequalities

$|L_{i}(\boldsymbol{x})|_{p}<pH^{-\tau_{i}}\,\,\text{ for }i=1,\dots,n.$

The following lemma generally states that the measure of a $\limsup$ set of
balls remains unaltered when the radius is multiplied by some constant. The
Euclidean version of this result is well known and appears in a variety of
texts, see [6]. The following version for ultrametric spaces was proven in
[11].

######  Lemma 3.2.

Let $(X,d)$ be a separable ultrametric space and $\mu$ be a Borel regular
measure on $X$. Let $(B_{i})_{i\in\mathbb{N}}$ be a sequence of balls in $X$
with radii $r_{i}\to 0$ as $i\to\infty$. Let $(U_{i})_{i\in\mathbb{N}}$ be a
sequence of $\mu$-measurable sets such that $U_{i}\subset B_{i}$ for all $i$.
Assume that for some $c>0$

$|U_{i}|\geq c|B_{i}|\qquad\text{for all }i\,.$

Then the limsup sets

$\textstyle\mathcal{U}=\limsup\limits_{i\to\infty}U_{i}:=\bigcap\limits_{j=1}^{\infty}\
\bigcup\limits_{i\geq j}U_{i}\qquad\text{ and
}\qquad\mathcal{B}=\limsup\limits_{i\to\infty}B_{i}:=\bigcap\limits_{j=1}^{\infty}\
\bigcup\limits_{i\geq j}B_{i}$

have the same $\mu$-measure.

In particular, if we chose the approximation function $\psi_{i}(q)=pq^{1+1/n}$
for each $1\leq i\leq n$ then by Lemma 3.1 we know
$\mathfrak{W}_{n}(\Psi)=\mathbb{Z}_{p}$, and so by shrinking the $\limsup$ set
of balls by constant $1/p$ Lemma 3.2 gives us that
$\mu_{n}(\mathfrak{W}_{n}(\Psi/p))=1$.

Another key result in our proof of Theorem 2.1 is the following Mass
Transference Principle type theorem. In order to state this theorem we need
the notion of local ubiquity for rectangles, a variation of the notion of
ubiquity introduced by Beresnevich, Dickinson, and Velani [6]. Fix an integer
$n\geq 1$, and for each $1\leq i\leq n$ let $(X_{i},|.|_{i},m_{i})$ be a
bounded locally compact metric space with $m_{i}$ a $\delta_{i}$-Ahlfors
probability measure. Consider the product space $(X,|.|,m)$, where

$X=\prod_{i=1}^{n}X_{i},\quad m=\prod_{i=1}^{n}m_{i},\quad|.|=\max_{1\leq
i\leq n}|.|_{i}.$

For any $x\in X$ and $r\in\mathbb{R}_{+}$ define the open ball

$B(x,r)=\left\\{y\in X:\max_{1\leq i\leq
n}|x_{i}-y_{i}|_{i}<r\right\\}=\prod_{i=1}^{n}B_{i}(x_{i},r),$

where $B_{i}$ are the usual balls associated with the $i^{\text{th}}$ metric
space. Let $J$ be a infinite countable index set, and
$\beta:J\to\mathbb{R}_{+}$ a positive function. Let $l_{k},u_{k}$ be two
sequences in $\mathbb{R}_{+}$ such that $u_{k}\geq l_{k}$ with
$l_{k}\to\infty$ as $k\to\infty$. Define

$J_{k}=\left\\{\alpha\in J:l_{k}\leq\beta_{\alpha}\leq u_{k}\right\\}.$

Let $\rho:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a non-increasing function with
$\rho(\beta_{\alpha})\to 0$ as $\beta_{\alpha}\to\infty$. For each $1\leq
i\leq n$, let $\\{R_{\alpha,i}\\}_{\alpha\in J}$ be a sequence of subsets in
$X_{i}$. As with the standard setting of ubiquitous systems define the
resonant sets

$\left\\{R_{\alpha}=\prod_{i=1}^{n}R_{\alpha,i}\right\\}_{\alpha\in J}.$

For $\boldsymbol{a}=(a_{1},\dots,a_{n})\in\mathbb{R}_{+}^{n}$ denote the set
of hyperrectangles

$\Delta(R_{\alpha},\rho(r)^{\boldsymbol{a}})=\prod_{i=1}^{n}\Delta(R_{\alpha,i},\rho(r)^{a_{i}}),$

where for some set $A$ and $b\in\mathbb{R}_{+}$

$\Delta(A,b)=\bigcup_{a\in A}B(a,b).$

######  Definition 3.3 (local ubiquitous system of rectangles).

Call $(\\{R_{\alpha}\\}_{\alpha\in J},\beta)$ a local ubiquitous system of
rectangles with respect to $(\rho,\boldsymbol{a})$ if there exists a constant
$c>0$ such that for any ball $B\subset X$,

$\limsup_{k\to\infty}m\left(B\cap\bigcup_{\alpha\in
J_{k}}\Delta(R_{\alpha},\rho(u_{k})^{\boldsymbol{a}})\right)\geq cm(B).$

The second property needed to state the theorem is the following local scaling
property, first introduced in [1]. While we will not need it in our use of the
Theorem 3.5 we state it and include it in the final theorem for completeness.

######  Definition 3.4 ($k$-scaling property).

Let $0\leq k<1$ and $1\leq i\leq n$. Then $\\{R_{\alpha,i}\\}_{\alpha\in J}$
has $k$-scaling property if for any $\alpha\in J$, any ball $B(x_{i},r)\subset
X_{i}$ with centre $x_{i}\in R_{\alpha,i}$, and $0<\epsilon<r$ then

$c_{2}r^{\delta_{i}k}\epsilon^{\delta_{i}(1-k)}\leq
m_{i}\left(B(x_{i},r)\cap\Delta(R_{\alpha,i},\epsilon)\right)\leq
c_{3}r^{\delta_{i}k}\epsilon^{\delta_{i}(1-k)},$

for some constants $c_{2},c_{3}>0$.

In our use the resonant sets will be sets of points, so $k=0$. For
$\textbf{t}=(t_{1},\dots,t_{n})\in\mathbb{R}^{n}_{+}$ define

$W(\textbf{t})=\limsup_{\alpha\in
J}\Delta\left(R_{\alpha},\rho(\beta_{\alpha})^{\boldsymbol{a}+\textbf{t}}\right).$

Given the above notations and definitions we can state the Mass Transference
Principle from rectangles to rectangles (MTPRR) of [41].

######  Theorem 3.5 (Mass Transference Principle from rectangles to
rectangles).

Under the settings above assume that $(\\{R_{\alpha}\\}_{\alpha\in J},\beta)$
satisfies the local ubiquity for systems of rectangles condition with respect
to $(\rho,\boldsymbol{a})$, and the $k$-scaling property. Then

$\dim W(\textbf{t})\geq\min_{A_{i}\in A}\left\\{\sum_{j\in
K_{1}}\delta_{j}+\sum_{j\in K_{2}}\delta_{j}+k\sum_{j\in
K_{3}}\delta_{j}+(1-k)\frac{\sum_{j\in K_{3}}a_{j}\delta_{j}-\sum_{j\in
K_{2}}t_{j}\delta_{j}}{A_{i}}\right\\}=s,$

where $A=\\{a_{i},a_{i}+t_{i},1\leq i\leq n\\}$ and $K_{1},K_{2},K_{3}$ are a
partition of $\\{1,\dots,n\\}$ defined as

$K_{1}=\\{j:a_{j}\geq A_{i}\\},\quad K_{2}=\\{j:a_{j}+t_{j}\leq
A_{i}\\}\backslash K_{1},\quad K_{3}=\\{1,\dots n\\}\backslash(K_{1}\cup
K_{2}).$

Further,

$\mathcal{H}^{s}(B\cap W(\boldsymbol{\tau}))=\mathcal{H}^{s}(B).$

Hence, provided we can find a $\limsup$ set of hyperrectangles that satisfy
the local ubiquity property for rectangles then we have a lower bound for the
corresponding $\limsup$ set of shrunken hyperrectangles.

##  4\. Proof of Theorem 2.1

We split the proof into the upper and lower bound, and solve each case
separately. In both cases we will use the following simplified set. Let $\pi$
be the projection $\pi:\mathbb{Z}_{p}^{n}\to\mathbb{Z}_{p}^{n-m}$, defined by

$(x_{1},\dots,x_{n})\mapsto(x_{1},\dots,x_{d}).$

By a well known theorem of Hausdorff theory (see Proposition 3.3 of [21]) as
$\pi$ is a bi-Lipschitz mapping over
$\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}$, we have
that

$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}=\dim\pi(\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}).$

Let $\boldsymbol{\tau}_{m}=(\tau_{d+1},\dots,\tau_{n})$ denote the $m$-tuple
of approximation exponents over $\boldsymbol{\alpha}$ and similarly let
$\boldsymbol{\tau}_{d}=(\tau_{1},\dots,\tau_{d})$ denote the $d$-tuple of
approximation exponents over the independent variables of
$\Pi_{\boldsymbol{\alpha}}$. Consider the set of integers

$\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m}):=\left\\{q_{0}\in\mathbb{N}:\left|\alpha_{i}-\frac{q_{d+i}}{q_{0}}\right|_{p}<q_{0}^{-\tau_{d+i}},\,\text{
for some }\,\begin{array}[]{c}|q_{i}|\leq q_{0},\\\ gcd(q_{i},q_{0})=1,\\\
\end{array}\,1\leq i\leq m\right\\},$

and the union of sets

$\mathfrak{A}_{q_{0}}^{*}(\boldsymbol{\tau}_{d})=\underset{1\leq i\leq
d}{\bigcup_{|q_{i}|\leq
q_{0},\,\,gcd(q_{i},q_{0})=1}}\left\\{\boldsymbol{x}\in\mathbb{Z}_{p}^{d}:\left|x_{i}-\frac{q_{i}}{q_{0}}\right|_{p}<q_{0}^{-\tau_{i}}\right\\}.$

Then,

$\pi(\mathcal{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}})=\limsup_{q_{0}\in\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m})}\mathfrak{A}^{*}_{q_{0}}(\boldsymbol{\tau}_{d}),$

hence we only need to find the upper and lower bounds for
$\dim\underset{q_{0}\in\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m})}{\limsup}\mathcal{A}^{*}_{q_{0}}(\boldsymbol{\tau}_{d})$.

###  4.1. Upper bound

For the upper bound we take the standard cover of hyperrectangles used in the
construction of $\mathfrak{A}^{*}_{q}(\boldsymbol{\tau}_{d})$. By a standard
geometrical argument note that each hyperrectangle, centred at some
$\left(\frac{q_{1}}{q},\dots,\frac{q_{n}}{q_{0}}\right)\in\mathbb{Q}^{d}$ in
the construction of $\mathfrak{A}_{q}^{*}(\boldsymbol{\tau}_{d})$, can be
covered by a finite collection of balls $\mathfrak{B}_{q}(\tau_{i})$ of radius
$q^{-\tau_{i}}$ for $1\leq i\leq d$. Without loss of generality we can assume
that

$\tau_{1}\geq\dots\geq\tau_{d},$

since if not then we could take some bi-Lipschitz mapping to reorder the
coordinate axes such that this was the case. Hence for each $j\leq i$,

$\frac{q^{-\tau_{j}}}{q^{-\tau_{i}}}\leq 1.$

Hence in the product below we only consider the $j\geq i$. By the above
argument we have that the cardinality of $\mathfrak{B}_{q}(\tau_{i})$ is

$\\#\mathfrak{B}_{q}(\tau_{i})\ll\prod_{j=i}^{d}\frac{q^{-\tau_{j}}}{q^{-\tau_{i}}}=q^{\sum_{j=i}^{d}(\tau_{i}-\tau_{j})}.$

As each $\tau_{i}$-approximation function is decreasing as $q$ increases, for
each interval $2^{k}\leq q<2^{k+1}$ take $q=2^{k}$ over such interval. Let

$\mathcal{Q}^{{}^{\prime}}(\boldsymbol{x},\boldsymbol{\tau}_{m},N):=\left\\{q_{0}\in\mathbb{N}:(q_{0},\dots,q_{m})\in\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau}_{m},N)\,\text{
and }gcd(q_{i},q_{0})=1\right\\}.$

Since each $\tau_{i}>1$ for $1\leq i\leq m$ each $q_{0}$ has unique associated
$(q_{1},\dots,q_{m})$ in $\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau}_{m},N)$
so we have that
$\\#\mathcal{Q}^{{}^{\prime}}(\boldsymbol{x},\boldsymbol{\tau}_{m},N)\leq\\#\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau}_{m},N)$.
Further, by the coprimality of each $q_{i}$ with $q_{0}$ note that the
inequalities

$|q_{0}x_{i}-q_{i}|_{p}<H^{-\tau_{i}},\quad\text{and}\quad\left|x_{i}-\frac{q_{i}}{q_{0}}\right|_{p}<H^{-\tau_{i}}$

are equivalent since $p\not|q_{0}$. To check this observe that each
$x_{i}\in\mathbb{Z}_{p}$ and then use the strong triangle inequality.

Given the above we have that
$\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m})\subseteq\bigcup_{k\in\mathbb{N}}\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},2^{k})$.
Hence

$\displaystyle\mathcal{H}^{s}\left(\limsup_{q\in\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m})}\mathcal{A}^{*}_{q}(\boldsymbol{\tau}_{d})\right)$
$\displaystyle\leq\sum_{k=1}^{\infty}\sum_{q\in\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},2^{k})}\phi(q)^{d}\\#\mathfrak{B}_{q}(\tau_{i}).(q^{-\tau_{i}})^{s},$
$\displaystyle\overset{(a)}{\ll}\sum_{k=1}^{\infty}2^{k(m+1-\sum_{i=1}^{m}\tau_{d+i})}(2^{k+1})^{d}(2^{k+1})^{\sum_{j=i}^{d}(\tau_{i}-\tau_{j})}(2^{k})^{-\tau_{i}s},$
$\displaystyle\ll\sum_{k=1}^{\infty}2^{k(n+1-\sum_{i=1}^{m}\tau_{d+i}+\sum_{j=i}^{d}(\tau_{i}-\tau_{j})-\tau_{i}s)},$

where $(a)$ follows from Theorem 1.2. The above sum converges when

$s\geq\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}+\sum_{j=i}^{d}(\tau_{i}-\tau_{j})}{\tau_{i}}+\epsilon,$

for any $\epsilon>0$. This is true for each $1\leq i\leq d$, and as $\epsilon$
is arbitrary, we have that

$s\geq\min_{1\leq i\leq
d}\left\\{\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}+\sum_{j=i}^{d}(\tau_{i}-\tau_{j})}{\tau_{i}}\right\\},$

completing the upper bound result. Note that the result of Remark 2.2 can
similarly be obtained by replacing Theorem 1.2 used at $(a)$ by Lemma 1.1.

###  4.2. Lower bound

In order to use Theorem 3.5 to prove the lower bound of Theorem 2.1 we need to
construct a ubiquitous system of rectangles. In following with the ubiquity
setup for Theorem 3.5 let

$\begin{array}[]{ccc}J=\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m}),&R_{q,i}=\left\\{\frac{q_{i}}{q}\in\mathbb{Q}:\begin{array}[]{c}|q_{i}|\leq
q,\\\ gcd(q_{i},q)=1\end{array}\right\\},&R_{q}=\prod_{i=1}^{d}R_{q,i},\\\
\beta(q)=q,&\rho(q)=q^{-1},&l_{k}=M^{k},\,\,u_{k}=M^{k+1},\end{array}$

where $M\in\mathbb{N}$ is a fixed integer to be determined later. Then we have
that

$J_{k}=\\{q\in\mathcal{Q}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m}):M^{k}\leq
q<M^{k+1}\\}.$

Note that
$J_{k}\subseteq\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},2^{k+1})$.
For a vector $\boldsymbol{a}=(a_{1},\dots,a_{n})\in\mathbb{R}^{n}_{+}$ let

$\Delta(R_{q},\rho(r)^{\boldsymbol{a}})=\prod_{i=1}^{n}\bigcup_{q_{i}\in
R_{q,i}}B\left(\frac{q_{i}}{q},r^{-a_{i}}\right).$

We prove the following.

######  Proposition 4.1.

Let $R_{q}$, $\rho$, and $J_{k}$ be as above, and let
$\tilde{v}=(v_{1},\dots,v_{d})\in\mathbb{R}^{d}_{>0}$ with each $v_{i}>1$ and

$\sum_{i=1}^{d}v_{i}=n+1-\sum_{i=1}^{m}\tau_{d+i},$

for $\sum_{i=1}^{m}\tau_{d+i}<m+1$ and each $\tau_{i}>1$. Then for any ball
$B=B(x,r)\subset\mathbb{Z}_{p}^{d}$, with centre $x\in\mathbb{Z}_{p}^{d}$ and
radius $0<r<r_{0}$ for some $r_{0}\in\mathbb{R}_{+}$, there exists a constant
$c>0$ such that

$\mu_{p,d}\left(B\cap\bigcup_{q\in
J_{k}}\Delta(R_{\boldsymbol{q}},\rho(u_{k})^{\tilde{v}})\right)\geq
c\mu_{p,d}(B),$

provided $M>(3^{d}C_{1})^{\frac{1}{n+1-\sum_{i=1}^{m}\tau_{d+i}}}$.

The proof of this result follows the same style of many similar results in
$\mathbb{R}^{n}$. For example see Theorem 1.3 of [12] for the one dimensional
real case, or Proposition 5.1 of [11] for the $n$-dimensional $p$-adic case.

###### Proof.

For any $y=(y_{1},\dots,y_{d})\in(\mathbb{Z}_{p}\backslash\mathbb{Q})^{d}$,
consider the system of inequalities

(2) $\begin{cases}|q_{0}\alpha_{i}-q_{d+i}|_{p}<(M^{k+1})^{-\tau_{d+i}},\quad
1\leq i\leq m,\\\ |q_{0}y_{i}-q_{i}|_{p}<(M^{k+1})^{-v_{i}},\quad 1\leq i\leq
d,\\\ \max_{0\leq i\leq n}|q_{i}|\leq M^{k+1}.\end{cases}$

By the condition on $\tilde{v}$ we have, by Lemma 3.1, that there exists a
non-zero integer solution $(q_{0},\dots,q_{n})\in\mathbb{Z}^{n+1}$ to (2).
Assume without loss of generality that $q_{0}\geq 0$. We prove that there
exists a rational integer solution $(q_{0},\dots,q_{n})$ to (2) satisfying

  1. i)

$q_{0}\neq 0$,

  2. ii)

$\frac{q_{i}}{q_{0}}\in\mathbb{Z}_{p}$ for each $1\leq i\leq n$,

  3. iii)

$gcd(q_{0},p)=1$.

Firstly, suppose $q_{0}=0$, then the above equations imply that each
$|q_{i}|_{p}<(M^{k+1})^{-\tau_{d+i}}$ for $1\leq i\leq m$ or
$|q_{i}|_{p}<(M^{k+1})^{-v_{i}}$ for $1\leq i\leq d$. However, since each
$\tau_{i}$ and $v_{i}$ are greater than one such inequalities are impossible,
unless $q_{i}=0$ for all $1\leq i\leq n$. But this solution is identically
zero.

For ii) note that since $q_{0}\neq 0$ we may divide each inequality in (2) by
$q_{0}$ to get

$\begin{cases}|q_{0}|_{p}\left|\alpha_{i}-\frac{q_{d+i}}{q_{0}}\right|_{p}<(M^{k+1})^{-\tau_{d+i}},\quad
1\leq i\leq m,\\\
|q_{0}|_{p}\left|y_{i}-\frac{q_{i}}{q_{0}}\right|_{p}<(M^{k+1})^{-v_{i}},\quad
1\leq i\leq d,\\\ \max_{0\leq i\leq n}|q_{i}|\leq M^{k+1}.\end{cases}$

Suppose $\left|\frac{q_{i}}{q_{0}}\right|_{p}>1$. Noting that each $y_{i}$ and
$\alpha_{i}$ are $p$-adic integers, and using the strong triangle inequality,
we obtain that each $|q_{i}|_{p}<(M^{k+1})^{-\tau_{i}}$ for $1\leq i\leq m$
and $|q_{i}|_{p}<(M^{k+1})^{-v_{i}}$ for $1\leq i\leq d$. As previously stated
such condition is impossible unless each $q_{i}=0$, in which case
$\left|\frac{q_{i}}{q_{0}}\right|_{p}=0$, contradicting that
$\left|\frac{q_{i}}{q_{0}}\right|_{p}>1$.

For iii) suppose that $(q_{0},\dots,q_{n})$ is a solution to (2) and suppose
that $p^{t}|q_{0}$ but $p^{t+1}\not|q_{0}$. Then by ii) we have that
$p^{t}|q_{i}$ for all $1\leq i\leq n$. Let $q_{i}^{\prime}=p^{-t}q_{i}$ for
each $0\leq i\leq n$. Note that

$\max_{0\leq i\leq
n}|q_{i}^{\prime}|\leq\frac{M^{k+1}}{p^{k}}=H^{{}^{\prime}}.$

Then for each $1\leq i\leq m$ we have that

$\displaystyle|q_{0}^{\prime}\alpha_{i}-q_{i}^{\prime}|_{p}$
$\displaystyle=p^{k}p^{-k}|q_{0}^{\prime}\alpha_{i}-q_{i}^{\prime}|_{p},$
$\displaystyle=p^{k}|q_{0}\alpha_{i}-q_{i}|_{p},$
$\displaystyle<p^{k}(M^{k+1})^{-\tau_{i}},$
$\displaystyle<(H^{{}^{\prime}})^{-\tau_{i}},$

and similarly for the approximations over $y$. Hence
$(q_{0}^{\prime},\dots,q_{n}^{\prime})$ is a rational integer solution with
$gcd(q_{0}^{\prime},p)=1$. Henceforth we will suppose $p\not|q_{0}$.

Since $gcd(q_{0},p)=1$ we may divide (2) through by $|q_{0}|_{p}=1$ and the
set of possible $y\in\mathbb{Z}_{p}^{d}$ remain unchanged. Lastly, note that
if $q_{0}$ has an associated solution to (2), then
$q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k+1})$,
thus we have that

$\mu_{p,d}\left(B\cap\bigcup_{q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k+1})}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right)=\mu_{p,d}(B)$

for any $B\subseteq\mathbb{Z}_{p}^{d}$. Note that

$\displaystyle\mu_{p,d}\left(B\cap\bigcup_{q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k+1})}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right)\leq\mu_{p,d}$
$\displaystyle\left(B\cap\bigcup_{q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k})}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right)$
$\displaystyle+\mu_{p,d}\left(B\cap\bigcup_{q_{0}\in
J_{k}}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right),$

and so

$\mu_{p,d}\left(B\cap\bigcup_{q_{0}\in
J_{k}}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right)\geq\mu_{p,d}(B)-\mu_{p,d}\left(B\cap\bigcup_{q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k})}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right).$

At this point we only want the
$\frac{\boldsymbol{q}}{q_{0}}=\left(\frac{q_{1}}{q_{0}},\dots,\frac{q_{d}}{q_{0}}\right)\in
R_{q_{0}}$ such that

$B\cap
B\left(\frac{\boldsymbol{q}}{q},\rho(M^{k+1})^{\tilde{v}}\right)\neq\emptyset.$

For ball $B=B(x,r)$ with $x\in\mathbb{Z}_{p}^{d}$ and
$r\in\\{p^{j}:j\in\mathbb{Z}\\}$, this is equivalent to the set of solutions
to

(3) $\left|x_{i}-\frac{q_{i}}{q_{0}}\right|_{p}<r,\quad 1\leq i\leq d.$

For $q_{0}$ fixed and each $|q_{i}|\leq q_{0}$ by congruence classes we have
that there are at most

$(2q_{0}r+1)^{d}$

suitable values of $\boldsymbol{q}$. We can choose suitably large
$k\in\mathbb{N}$ such that $M^{k}r>1$, and so for each $|q_{i}|\leq M^{k}$,
$1\leq i\leq d$ there are at most

(4) $(3M^{k}r)^{d}$

possible values of $\boldsymbol{q}$ solving (3). Hence

$\displaystyle\mu_{p,d}\left(B\cap\bigcup_{q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k})}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right)$
$\displaystyle\leq\sum_{q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k})}\sum_{\boldsymbol{q}\,\,\text{solving}\,\,\eqref{system2}}\mu_{p,d}\left(B\cap\Delta\left(\frac{\boldsymbol{q}}{q_{0}},\rho(M^{k+1})^{\tilde{v}})\right)\right),$
$\displaystyle\overset{(a)}{\leq}\sum_{q_{0}\in\mathcal{Q}^{{}^{\prime}}(\boldsymbol{\alpha},\boldsymbol{\tau}_{m},M^{k})}(3M^{k}r)^{d}(M^{k+1})^{-d\sum_{i=1}^{d}v_{i}},$
$\displaystyle\overset{(b)}{\leq}C_{1}M^{k(m+1-\sum_{i=1}^{m}\tau_{d+i})}3^{d}M^{kd}M^{-(k+1)(n+1-\sum_{i=1}^{m}\tau_{d+i}}\mu_{p,d}(B),$
$\displaystyle\leq 3^{d}C_{1}M^{-n-1+\sum_{i=1}^{m}\tau_{d+i}}\mu_{p,d}(B),$

where $(a)$ follows by (4) and $(b)$ follows by Theorem 1.2 and our condition
on $\tilde{v}$. As $M>(3^{d}C_{1})^{\frac{1}{n+1-\sum_{i=1}^{m}\tau_{d+i}}}$,

$c=\left(1-\frac{3^{d}C_{1}}{M^{n+1-\sum_{i=1}^{m}\tau_{d+i}}}\right)>0.$

Thus,

$\mu_{p,d}\left(B\cap\bigcup_{q_{0}\in
J_{k}}\Delta(R_{q_{0}},\rho(M^{k+1})^{\tilde{v}}))\right)\geq c.\mu_{d}(B).$

∎

Given Proposition 4.1 we have that $(R_{q},\beta)$ is a local ubiquitous
system of rectangles with respect to $(\rho,\tilde{v})$, provided
$\sum_{i=1}^{d}v_{i}=n+1-\sum_{i=1}^{m}\tau_{d+i}$. Given
$\boldsymbol{\tau}_{d}=(\tau_{1},\dots,\tau_{d})\in\mathbb{R}^{d}_{>0}$ assume
without loss of generality that $\tau_{1}>\tau_{2}>\dots>\tau_{d}$ and define
each $v_{d-i}$ recursively by

$v_{d-i}=\min\left\\{\tau_{d-i},\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}-\sum_{j=d-i+1}^{d}v_{j}}{d-i}\right\\}.$

By the condition on $\boldsymbol{\tau}_{d}$ of Theorem 2.1, there exists a
$k\in\\{1,\dots,d\\}$ such that

$v_{l}=\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}-\sum_{j=d-k+1}^{d}v_{j}}{d-k},$

for all $1\leq l\leq d-k$. Clearly each $v_{i}\leq\tau_{i}$ for $1\leq i\leq
d$, and so the associated vector $\mathbf{t}=(t_{1},\dots
t_{n-1})\in\mathbb{R}^{n-1}_{\geq 0}$ is defined by

$t_{i}=\tau_{i}-v_{i},\quad 1\leq i\leq d.$

Consider the set

$A=\\{v_{1},\dots,v_{d},\tau_{1},\dots,\tau_{d}\\}.$

For each $A_{i}\in A$ observe the following:

  1. i)

$A_{i}\in\\{v_{1},\dots,v_{d}\\}$: Then we have the sets

$\begin{array}[]{ccc}K_{1}=\\{1,\dots,\max\\{i,d-k\\}\\},&K_{2}=\\{\max\\{i+1,d-k+1\\},\dots,d\\},&K_{3}=\emptyset.\end{array}$

By Theorem 3.5 we have that

$\displaystyle\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}$
$\displaystyle\geq\min_{A_{i}}\left\\{\frac{\max\\{i,d-k\\}v_{i}+(d-\max\\{i+1,d-k+1\\})v_{i}-\sum_{j=\max\\{i+1,d-k+1\\}}^{d}t_{j}}{v_{i}}\right\\},$
$\displaystyle=\min_{A_{i}}\left\\{\frac{dv_{i}-\sum_{j=\max\\{i+1,d-k+1\\}}^{d}t_{j}}{v_{i}}\right\\}.$

Since $t_{j}=0$ for $d-k+1\leq j\leq d$ the above equation gives that
$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})=d=n-m$, the maximal dimension of
$\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}$.

  2. ii)

$A_{i}\in\\{\tau_{1},\dots,\tau_{d}\\}$: Since $\tau_{i}=v_{i}$ for $d-k+1\leq
i\leq d$ the above argument covers such case, so we only need to consider
$\tau_{i}$ for $1\leq i\leq d-k$. For such $\tau_{i}$ we have the sets

$\begin{array}[]{ccc}K_{1}=\emptyset,&K_{2}=\\{i,\dots,d\\},&K_{3}=\\{1,\dots,i-1\\}.\end{array}$

Applying Theorem 3.5 we have

$\displaystyle\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}$
$\displaystyle\geq\min_{A_{i}}\left\\{\frac{(d-i)\tau_{i}+\sum_{j=1}^{i-1}v_{j}-\sum_{j=i}^{d}t_{j}}{\tau_{i}}\right\\},$
$\displaystyle=\min_{A_{i}}\left\\{\frac{(d-i)\tau_{i}+(d-k)\left(\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}-\sum_{j=d-k+1}^{d}v_{j}}{d-k}\right)-\sum_{j=1}^{d-k}\tau_{j}}{\tau_{i}}\right\\},$
$\displaystyle=\min_{A_{i}}\left\\{\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}+\sum_{j=i}^{d}(\tau_{i}-\tau_{j})}{\tau_{i}}\right\\}.$

Combining i) and ii) we have that

$\dim\mathfrak{W}_{n}(\boldsymbol{\tau})\cap\Pi_{\boldsymbol{\alpha}}\geq\min_{1\leq
i\leq
d}\left\\{\frac{n+1-\sum_{i=1}^{m}\tau_{d+i}+\sum_{j=i}^{d}(\tau_{i}-\tau_{j})}{\tau_{i}}\right\\},$

completing the proof.

##  5\. Proof of the counting results

Recall, we are aim to provide bounds on the set

$\mathcal{Q}(\boldsymbol{x},\Psi,N):=\left\\{(q_{0},\dots,q_{n})\in\mathbb{Z}^{n+1}:\begin{array}[]{c}0<q_{0}\leq
N,\\\ \max_{1\leq i\leq n}|q_{i}|\leq N,\\\
\end{array}\,\left|q_{0}x_{i}-q_{i}\right|_{p}<\psi_{i}(N),\,1\leq i\leq
n\right\\}.$

We begin with the proof of Lemma 1.3. This style of proof is not new and
follows a similar method to the proof in the euclidean case (see Lemma 3 of
[36]).

Proof of Lemma 1.3: Fix
$\boldsymbol{x}=(x_{1},\dots,x_{n})\in\mathbb{Z}_{p}^{n}$ and take
$t=(t_{1},\dots,t_{n})\in\mathbb{N}^{n}$ to be the integers such that

$p^{-t_{i}}\leq N^{-\tau_{i}}<p^{-t_{i}+1},\quad 1\leq i\leq n.$

Denote by $P=\prod_{i=1}^{n}p^{t_{i}}$. Consider a set of open disjoint
rectangles $\\{R_{i}\\}_{i=1}^{P}$, each with some centre point
$k_{i}=(k_{i,1},\dots,k_{i,n})\in\mathbb{Z}^{n}$ and sidelenghts $p^{-t_{i}}$.
Choose the set of points $\\{k_{i}\\}$ such that
$\mathbb{Z}_{p}^{n}\subseteq\bigcup_{i=1}^{P}R_{i}$. Consider the
$(N+1)^{n+1}$ set of points of the form

$(q_{0}x-q)=(q_{0}x_{1}-q_{1},\dots,q_{0}x_{n}-q_{n})\in\mathbb{Z}_{p}^{n},$

with $q_{i}\in[0,N]$ for each $0\leq i\leq n$. By the Pigeon-hole principle
there exists at least one rectangle, say $R_{j}$, containing at least

$\frac{(N+1)^{n+1}}{P}>\frac{1}{p^{n}}N^{n+1-\sum_{i=1}^{n}\tau_{i}}$

points. As $\sum_{i=1}^{n}\tau_{i}<n+1$ we can choose $N$ sufficiently large
enough such that $p^{-n}N^{n+1-\sum_{i=1}^{n}\tau_{i}}>2$. Order the points
$(q_{0},\dots,q_{n})$, correspond to the points $q_{0}x-q$ contained in
$R_{j}$, by the absolute value of the $q_{0}$ component. If the $q_{0}$
components are equal then order by $q_{1}$ and so on. Suppose that the vector
$(m_{0},\dots,m_{n})$ is the smallest by our ordering. Then for all other
vectors $(r_{0},\dots,r_{n})$ contained in $R_{j}$ we have that

$\displaystyle|k_{j,i}-(m_{0}x_{i}-m_{i})-(k_{j,i}-(r_{0}x_{i}-r_{i})|_{p}$
$\displaystyle<p^{-t_{i}},$
$\displaystyle|(r_{0}-m_{0})x_{i}-(r_{i}-m_{i})|_{p}$
$\displaystyle<p^{-t_{i}}\leq N^{-\tau_{i}}.$

Hence the vectors $(r_{0}-m_{0},\dots,r_{n}-m_{n})\in\mathbb{Z}^{n+1}$ solve
the inequality of $\mathcal{Q}(\boldsymbol{x},\boldsymbol{\tau},N)$. Further
$(r_{i}-m_{i})\in[-N,N]$, and by the ordering stated above
$r_{0}-m_{0}\in[0,N]$. To exclude the case where $r_{0}-m_{0}=0$ observe that
each $\tau_{i}>1$ and so we would have that

$N^{-1}\leq|r_{i}-m_{i}|_{p}<p^{-t_{i}}<N^{-1}$

for $1\leq i\leq n$, a contradiction. The above argument yields
$p^{-n}N^{n+1-\sum_{i=1}^{n}\tau_{i}}-1$ such points, completing the proof. ∎

Lemma 1.1 is also a relatively simple proof. We are unable to find a proof
that uses a similar argument, however we suspect such style of result has been
used before.

Proof of Lemma 1.1: We use a proof by contradiction. Suppose that

(5) $\\#\mathcal{Q}(x,\tau,N)>2N^{\tau(x)-\tau+\epsilon}.$

We use the following notations. Let $X\in\mathbb{N}$ be an integer such that

$|x-X|_{p}<p^{-M},$

for some suitably large $M\in\mathbb{N}$. Define $V_{N}^{+}$ and $V_{N}^{-}$
to be the sets

$\displaystyle V_{N}^{+}:=\\{(q,q_{1})\in\mathbb{N}\times\mathbb{Z}:0<q\leq
N,\,0\leq q_{1}\leq N,\,\\},$ $\displaystyle
V_{N}^{-}:=\\{(q,q_{1})\in\mathbb{N}\times\mathbb{Z}:0<q\leq N,\,-N\leq
q_{1}\leq 0,\,\\}.$

Let $t\in\mathbb{N}$ be the integer such that

$p^{-t}\leq N^{-\tau}<p^{-t+1},$

and similarly $k\in\mathbb{N}$ be the integer such that

$p^{-k}\leq N^{-(\tau(x)+\epsilon)}<p^{-k+1}.$

Note that as $\tau(x)>\tau$, we have that $k\geq t$, and so
$p^{k-t}\in\mathbb{N}$. Further, observe that

(6) $p^{k-t}<pN^{\tau(x)-\tau+\epsilon}.$

Lastly, by the definition of $\tau(x)$, we have that there exists finitely
many $Q\in\mathbb{N}$ such that

$|qx-q_{1}|_{p}<Q^{-(\tau(x)+\epsilon)},$

for $0<q,|q_{1}|\leq Q$. Hence our ’sufficiently large $N_{0}$’ is the value
of $N_{0}$ such that for any pair $0<q,|q_{1}|\leq N$,

(7) $|qx-q_{1}|_{p}\geq N^{-(\tau(x)+\epsilon)},$

for all $\epsilon>0$. Consider the set of points in $\mathcal{Q}(x,\tau,N)$.
Note that $(q,q_{1})\in\mathcal{Q}(x,\tau,N)$ if and only if $(q,q_{1})\in
V_{N}^{+}\cup V_{N}^{-}$, and

(8) $qX-q_{1}\equiv 0\mod p^{t}.$

Thus, for all $(q,q_{1})\in\mathcal{Q}(x,\tau,N)$ we have that

$qX-q_{1}=\lambda p^{t},$

for some $\lambda\in\mathbb{Z}$. Split the set of points in
$\mathcal{Q}(x,\tau,N)$ into two disjoint sets, the set of pairs in
$V_{N}^{+}$, and the set of pairs in $V_{N}^{-}$. As there are greater than
$2N^{\tau(x)-\tau+\epsilon}$ pairs, at least one of the sets has greater than
$N^{\tau(x)-\tau+\epsilon}$ pairs. Without loss of generality assume such set
of points belong in $V_{N}^{+}$. Considering the range of values of $\lambda
p^{t}$ there are $p^{k-t}$ possible values of $\lambda p^{t}$ modulo $p^{k}$.
By (5) and (6) we have, by the Pigeon-hole principle, that there exists at
least two pairs, say $(a,a_{1})$ and $(b,b_{1})$, such that

$(a-b)X-(a_{1}-b_{1})\equiv 0\mod p^{k}.$

This is equivalent to

$|(a-b)x-(a_{1}-b_{1})|_{p}\leq p^{-k}\leq N^{-(\tau(x)+\epsilon)},$

with $(a-b,a_{1}-b_{1})\in V_{N}^{+}\cup V_{N}^{-}$, as $0<a-b\leq N$ by our
choice of ordering of $a,b$, and $|a_{1}-b_{1}|\leq N$ by the fact that the
pairs $(a,a_{1}),(b,b_{1})\in V_{N}^{+}$. However, such result contradicts (7)
which follows from the definition of $\tau(x)$, thus (5) must be false. ∎

###  5.1. $p$-adic approximation lattices

Prior to the proof of Theorem 1.2 we recall some basic definitions and results
of Lattice theory that will be needed. Define a lattice $\Lambda$ as a
discrete additive subgroup of $\mathbb{R}^{n}$. If
$\Lambda\subseteq\mathbb{Z}^{n}$ the $\Lambda$ is an integer lattice. A set of
linearly independent vectors $b_{1},\dots,b_{n}$ that generate $\Lambda$ is
called a basis of $\Lambda$. Let $B$ be a $n\times n$ matrix with columns
$b_{i}$, then call $B$ a basis matrix. Define the fundamental region as

$\mathcal{F}(B):=\left\\{\sum_{i=1}^{n}a_{i}b_{i}:a_{i}\in\mathbb{R},\,\,0\leq
a_{i}<1\right\\}.$

A standard result of Lattice theory states that if $B$ is a basis matrix for
$\Lambda$ then $\mathcal{F}(B)$ contains no lattice points other than the
origin (see Chapter 3, Lemma 6 of [18]).

The volume of the fundamental region can be found by taking the determinant of
the basis matrix, that is $vol(\mathcal{F}(B))=|\det B|$. A basis matrix is
not unique for each $\Lambda$, however for any lattice $\Lambda$ the volume of
the fundamental region is the same regardless of choice of basis matrix. For
this reason the notation $vol(\mathcal{F}(B))=|\det\Lambda|$ is used. If
$U\in\mathbb{Z}^{n\times n}$ is a unimodular matrix and $B_{1}$ is a basis
matrix for $\Lambda$ then $B_{2}=B_{1}U$ is also a basis matrix for $\Lambda$.

One property of lattices that are incredibly useful are the successive minima
of a lattice. Let $B_{n}=B(0,1)$ denote the $n$-dimensional unit sphere. For
$c\in\mathbb{R}_{+}$ we use the notation $cB_{n}=B(0,c)$. Define the
successive minima of a lattice $\Lambda\subset\mathbb{R}^{n}$ of rank $n$ as
the set of values

$\lambda_{i}(\Lambda):=\min\\{\lambda>0:\dim(\Lambda\cap\lambda B)\geq i\\},$

for $i=1,\dots,n$. By Minkowski’s inequalities on the successive minima (see
e.g. [24]) we have that

(9) $vol(B_{n})\prod_{i=1}^{n}\lambda_{i}(\Lambda)\leq 2^{n}\det\Lambda.$

For a count on the number of lattice points within a convex body we have the
follow theorem due to Blichfeldt [15].

######  Theorem 5.1.

Let $\Lambda\subset\mathbb{R}^{n}$ be a lattice of rank $n$ and let
$V\subset\mathbb{R}^{n}$ be a convex body such that $rank(\Lambda\cap V)=n$.
Then

$\\#(\Lambda\cap V)\leq n!\frac{vol(V)}{\det\Lambda}+n.$

The constant for such estimate can be excessively large, however in our use of
the Theorem the size of such constant is irrelevant.

In 1993 an alternative lattice counting theorem was proven by Betke, Henk and
Wills [14], which utilised the properties of the successive minima. This
result was further generalised by Henk [23], giving us the following theorem.

######  Theorem 5.2.

Let $n\geq 2$, $B(0,K)$ a $n$-dimensional ball of radius $K>0$ centred at the
origin and $\Lambda$ a $n$-dimensional lattice. Then

$\\#(\Lambda\cap
K)<2^{n-1}\prod_{i=1}^{n}\left\lfloor\frac{2K}{\lambda_{i}(\Lambda)}+1\right\rfloor.$

We remark that if $rank(\Lambda\cap B(0,M))<n$ then we must have at least that
$\lambda_{n}(\Lambda)\geq M$. Thus the $n^{th}$ value of the product in
Theorem 5.2 would be bounded above by $3$, a point we make use of later on.

For the proof of Theorem 1.2 we use $p$-adic approximation lattices. First
discovered by de Weger [20] who used them to prove a variety of results in
classical $p$-adic Diophantine approximation, including the $p$-adic analogue
of Hurwitz Theorem. Recently $n$-dimensional forms of $p$-adic approximation
lattices have been used to provided lattice based cryptosystems [28, 29]. In
these papers both dual and simultaneous approximation lattices were discussed.
In particular Dirichlet-style exponents were proven for simultaneous and dual
approximation.

For a $n$-tuple of approximation functions $\Psi=(\psi_{1},\dots,\psi_{n})$,
an integer $N\in\mathbb{N}$, and a fixed
$\boldsymbol{x}=(x_{1},\dots,x_{n})\in\mathbb{Z}_{p}^{n}$ define the
$\Psi$-approximation lattice $\Lambda_{N,\boldsymbol{x}}$ by

$\Lambda_{N,\boldsymbol{x}}=\\{(a_{0},\dots,a_{n})\in\mathbb{Z}^{n+1}:|a_{0}x_{i}-a_{i}|_{p}\leq\psi_{i}(N),\,\,1\leq
i\leq n\\}.$

Observe that

$\mathcal{Q}(\boldsymbol{x},\Psi,N)\subseteq\Lambda_{N,\boldsymbol{x}}\cap
B(0,\sqrt{n}N),$

since the euclidean ball $B(0,\sqrt{n}N)$ contains all integer points
satisfying $\max_{0\leq i\leq n}|q_{i}|\leq N$.

For any $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$ we may write each $x_{j}$ as the
$p$-adic expansion

$x_{j}=\sum_{i=0}^{\infty}x_{j,i}p^{i},\quad x_{j,i}\in\\{0,1,\dots,p-1\\}.$

Let $X_{j,N}\in\mathbb{Z}$ be the integer

$X_{j,N}=\sum_{i=0}^{t_{j}}x_{j,i}p^{i},$

where each $t_{j}\in\mathbb{N}$ is the unique value associated with $N$
satisfying

(10) $p^{-t_{j}}\leq\psi_{j}(N)<p^{-t_{j}+1}.$

Lastly, for each $1\leq j\leq n$ let $\psi_{j,N}^{*}=p^{t_{j}}$. Then the set
of vectors

(11) $B=\left\\{\left(\begin{array}[]{c}1\\\ X_{1,N}\\\ \vdots\\\
X_{n,N}\end{array}\right),\left(\begin{array}[]{c}0\\\ \psi_{1,N}^{*}\\\
\vdots\\\ 0\end{array}\right),\dots,\left(\begin{array}[]{c}0\\\ 0\\\
\vdots\\\ \psi_{n,N}^{*}\end{array}\right)\right\\},$

form a basis for $\Lambda_{N,\boldsymbol{x}}$. The set $B$ can be proven to be
a basis by considering the fundamental region $\mathcal{F}(B)$ and showing the
only lattice point contained is 0. Given such basis we can deduce that

$|\det\Lambda_{N,\boldsymbol{x}}|=\prod_{i=1}^{n}\psi_{i,N}^{*}\asymp\left(\prod_{i=1}^{n}\psi_{i}(N)\right)^{-1},$

where the implied constants can be easily found using (10) to obtain

(12)
$\left(\prod_{i=1}^{n}\psi_{i}(M)\right)^{-1}\leq|\det\Lambda_{N,\boldsymbol{x}}|\leq
p^{n}\left(\prod_{i=1}^{n}\psi_{i}(N)\right)^{-1}.$

In the simultaneous case, $\Psi=(\psi,\dots,\psi)$, it was proven in [29] that

$\lambda_{1}(\Lambda_{N,\boldsymbol{x}})\ll\psi(N)^{-\frac{n}{n+1}}.$

In order to prove Theorem 1.2 we find a lower bound on
$\lambda_{1}(\Lambda_{N,\boldsymbol{x}})$ by considering
$\boldsymbol{x}\in\mathbb{Z}_{p}$ satisfying certain Diophantine exponent
properties. For completeness we also prove the upper bound.

######  Lemma 5.3.

Let $\Lambda_{N,\boldsymbol{x}}$ be defined above with
$\tau(\boldsymbol{x})=n+1$, and suppose that

$\prod_{i=1}^{n}\psi_{i}(N)<N^{-n}.$

Then for any $\varepsilon>0$ the exists sufficiently large
$N_{0}\in\mathbb{N}$ such that for all $N\geq N_{0}$,

$\left(\frac{1}{\prod_{i=1}^{n}\psi_{i}(N)}\right)^{\frac{1}{n+1}-\varepsilon}\leq\lambda_{1}(\Lambda_{N,\boldsymbol{x}})\leq
C_{2}\left(\frac{1}{\prod_{i=1}^{n}\psi_{i}(N)}\right)^{\frac{1}{n+1}},$

where

$C_{2}=2\left(\frac{\Gamma\left(\frac{n+1}{2}+1\right)p^{n}}{\pi^{\frac{n+1}{2}}}\right)^{\frac{1}{n+1}}.$

As will become clear in the proof below the condition that
$\tau(\boldsymbol{x})=n+1$ is only necessary in the lower bound result.

###### Proof.

We prove the upper bound case first. Such proof is a standard application of
Minkowski’s first Theorem on successive minima and follows almost immediately
by the above calculation of $\det(\Lambda_{M,\boldsymbol{x}})$. Concisely, we
have that

$\lambda_{1}(\Lambda_{N,\boldsymbol{x}})^{n+1}vol(B(0,1))\leq
2^{n+1}\det(\Lambda_{N,\boldsymbol{x}}).$

Rearranging for $\lambda_{1}(\Lambda_{N,\boldsymbol{x}})$, using (12), and
recalling the volume of an $n+1$-ball we obtain our result.

We now prove the lower bound. For any $\boldsymbol{x}\in\mathbb{Z}_{p}^{n}$
observe that

$\prod_{i=1}^{n}|q_{0}x_{i}-q_{i}|_{p}<M^{-(n+1)},$

for infinitely many $M$ (see for example Lemma 3.1). Further, since
$\tau(\boldsymbol{x})=n+1$ there exists $N_{0}$ such that for all $N\geq
N_{0}$ then any rational integer vectors $(q_{0},\dots,q_{n})$ satisfying
$\max_{0\leq i\leq n}|q_{i}|\leq N$ we have that

(13) $\prod_{i=1}^{n}|q_{0}x_{i}-q_{i}|_{p}\geq N^{-(n+1+\varepsilon)}$

for some $\varepsilon>0$. Choose $N$ sufficiently large such that

$N_{0}\leq\left(\prod_{i=1}^{n}\psi_{i}(N)\right)^{-\left(\frac{1}{n+1}-\varepsilon\right)}.$

Such $N$ is possible since $\prod_{i=1}^{n}\psi_{i}(N)<N^{-n}$ and so the
value on the RHS of the above inequality tends to infinity as $N\to\infty$ for
any small $\varepsilon$ ($\varepsilon<\frac{1}{n(n+1)}$).

Suppose that $(q_{0},\dots,q_{n})$ is a minimum length vector of
$\Lambda_{N,\boldsymbol{x}}$, then note that
$\lambda_{1}(\Lambda_{N,\boldsymbol{x}})\geq\max_{1\leq i\leq n}|q_{i}|$ due
to the euclidean nature of $\lambda_{1}(\Lambda_{N,\boldsymbol{x}})$. Suppose
that

(14) $\max_{1\leq i\leq
n}|q_{i}|<\left(\prod_{i=1}^{n}\psi_{i}(N)\right)^{-\left(\frac{1}{n+1}-\varepsilon\right)}.$

We prove (14) to be false. Observe that

$\prod_{i=1}^{n}|q_{0}x_{i}-q_{i}|_{p}<\prod_{i=1}^{n}\psi_{i}(N),$

since $(q_{0},\dots,q_{n})\in\Lambda_{N,\boldsymbol{x}}$. Then

$\prod_{i=1}^{n}|q_{0}x_{i}-q_{i}|_{p}<\left(\left(\prod_{i=1}^{n}\psi_{i}(N)\right)^{-\left(\frac{1}{n+1}-\varepsilon\right)}\right)^{-\frac{n+1}{1-\varepsilon(n+1)}}.$

But this contradicts (13). So we must have that (14) is false, and so

$\lambda_{1}(\Lambda_{N,\boldsymbol{x}})\geq\left(\prod_{i=1}^{n}\psi_{i}(N)\right)^{-\left(\frac{1}{n+1}-\varepsilon\right)},$

completing the proof. ∎

Given Lemma 5.3 we can proceed with the following.
Proof of Theorem 1.2: For $N\geq N_{0}$, where $N_{0}$ is chosen by Lemma 5.3,
consider the following two cases:

  1. i)

$rank(\Lambda_{N,\boldsymbol{x}}\cap B(0,\sqrt{n}N))=n+1$: By Theorem 5.1 we
have that

$\displaystyle\\#(\Lambda_{N,\boldsymbol{x}}\cap B(0,\sqrt{n}N))$
$\displaystyle\leq(n+1)!\frac{vol(B(0,\sqrt{n}N))}{\det\Lambda_{N,\boldsymbol{x}}}+n+1,$
$\displaystyle\leq\frac{(n+1)!\pi^{n/2}\sqrt{n}^{n+1}}{\Gamma\left(\frac{n}{2}+1\right)}N^{n+1}.(\prod_{i=1}^{n}\psi_{i,N}^{*})^{-1}+n+1,$
$\displaystyle\leq\frac{(n+2)!\pi^{n/2}\sqrt{n}^{n+1}}{\Gamma\left(\frac{n}{2}+1\right)}N^{n+1}\prod_{i=1}^{n}\psi_{i}(N).$

Note that the last inequality follows since
$\prod_{i=1}^{n}\psi_{i}(N)>N^{-(n+1-\varepsilon)}$. This proves Theorem 1.2
for the rank $n+1$ case.

  2. ii)

$rank(\Lambda_{N,\boldsymbol{x}}\cap B(0,\sqrt{n}N))<n+1$: Since
$rank(\Lambda_{N,\boldsymbol{x}}\cap B(0,\sqrt{n}N))<n+1$ we must have
$\lambda_{n+1}(\Lambda_{N,\boldsymbol{x}})>\sqrt{n}N$. Hence, by the remark
made previously, the final product on the right of Theorem 5.2 is less than or
equal to $3$. Furthermore, for each $\lambda_{i}(\Lambda_{N,\boldsymbol{x}})$,
$1\leq i\leq n$ we have that

$\displaystyle\lambda_{n}(\Lambda_{N,\boldsymbol{x}})\geq\dots\geq\lambda_{1}(\Lambda_{N,\boldsymbol{x}})$
$\displaystyle\overset{(a)}{\geq}\left(\frac{1}{\prod_{i=1}^{n}\psi_{i}(N)}\right)^{\frac{1}{n+1}-\varepsilon},$
$\displaystyle\overset{(b)}{\geq}\left(\frac{1}{N\prod_{i=1}^{n}\psi_{i}(N)}\right)^{1/n},$

where $(a)$ follows from Lemma 5.3 and $(b)$ follows since

$\displaystyle\left(\frac{1}{\prod_{i=1}^{n}\psi_{i}(N)}\right)^{\frac{1}{n+1}-\varepsilon}$
$\displaystyle\geq\left(\left(\frac{1}{\prod_{i=1}^{n}\psi_{i}(N)}\right)^{n-\varepsilon
n(n+1)}\right)^{\frac{1}{n(n+1)}},$
$\displaystyle\geq\left(\frac{1}{N^{n+1}\prod_{i=1}^{n}\psi_{i}(N)}\left(\frac{1}{\prod_{i=1}^{n}\psi_{i}(N)}\right)^{n}\right)^{\frac{1}{n(n+1)}}.$

combining the two ideas above, and Theorem 5.2, we have that

$\displaystyle\\#(\Lambda_{N,\boldsymbol{x}}\cap B(0,\sqrt{n}N))$
$\displaystyle<2^{n}3\prod_{i=1}^{n}\left(\frac{2\sqrt{n}N}{\lambda_{1}(\Lambda_{N,\boldsymbol{x}})}+1\right),$
$\displaystyle<2^{n}3\left(2\sqrt{n}N^{1+1/n}\left(\prod_{i=1}^{n}\psi_{i}(N)\right)^{1/n}+1\right)^{n},$
$\displaystyle<3(6\sqrt{n})^{n}N^{n+1}\prod_{i=1}^{n}\psi_{i}(N).$

Thus, in either case $i)$ or $ii)$ we have that

$\\#(\Lambda_{N,\boldsymbol{x}}\cap B(0,\sqrt{n}N))\leq
C_{1}N^{n+1}\prod_{i=1}^{n}\psi_{i}(N),$

with

$C_{1}=\max\left\\{3(6\sqrt{n})^{n},\frac{(n+2)!\pi^{n/2}\sqrt{n}^{n+1}}{\Gamma\left(\frac{n}{2}+1\right)}\right\\}.$

∎

##  6\. Concluding remarks

This article provides sharp bounds on the number of rational points close to
almost all $p$-adic integers. While this result allows us to find simultaneous
$p$-adic Diophantine approximation results on coordinate hyperplanes, it falls
a long way short of providing results for Diophantine approximation sets on
curves and manifolds. It is hoped the techniques used in this paper could be
used to find rational points close to manifolds, we intend to follow this up
with a subsequent paper.

## References

  * [1] D Allen and S Baker.  A general mass transference principle.  Selecta Math. (N.S.), 25(3):Paper No. 39, 38, 2019.
  * [2] D Badziahin and Y Bugeaud.  On simultaneous rational approximation to a real number and its integral powers, II.  New York J. Math., 26:362–377, 2020.
  * [3] D Badziahin, Y Bugeaud, and J Schleischitz.  On simultaneous rational approximation to a $p$-adic number and its integral powers, ii.  https://arxiv.org/abs/1511.06862.
  * [4] V Beresnevich.  Rational points near manifolds and metric Diophantine approximation.  Ann. of Math. (2), 175(1):187–235, 2012.
  * [5] V Beresnevich, V Bernik, and E Kovalevskaya.  On approximation of $p$-adic numbers by $p$-adic algebraic numbers.  J. Number Theory, 111(1):33–56, 2005.
  * [6] V Beresnevich, D Dickinson, and S Velani.  Measure theoretic laws for lim sup sets.  Mem. Amer. Math. Soc., 179(846):x+91, 2006.
  * [7] V Beresnevich, D Dickinson, and S Velani.  Diophantine approximation on planar curves and the distribution of rational points.  Ann. of Math. (2), 166(2):367–426, 2007.  With an Appendix II by R. C. Vaughan.
  * [8] V Beresnevich, A Haynes, and S Velani.  Sums of reciprocals of fractional parts and multiplicative Diophantine approximation.  Mem. Amer. Math. Soc., 263(1276):vii + 77, 2020.
  * [9] V Beresnevich and È Kovalevskaya.  On Diophantine approximations of dependent quantities in the $p$-adic case.  Mat. Zametki, 73(1):22–37, 2003.
  * [10] V Beresnevich, L Lee, R.C Vaughan, and S Velani.  Diophantine approximation on manifolds and lower bounds for Hausdorff dimension.  Mathematika, 63(3):762–779, 2017.
  * [11] V Beresnevich, J Levesley, and B Ward.  Simultaneous $p$-adic Diophantine approximation.  https://arxiv.org/abs/2101.05251.
  * [12] V Beresnevich, F Ramírez, and S Velani.  Metric Diophantine approximation: aspects of recent work.  In Dynamics and analytic number theory, volume 437 of London Math. Soc. Lecture Note Ser., pages 1–95. Cambridge Univ. Press, Cambridge, 2016.
  * [13] V Beresnevich, R. C. Vaughan, S Velani, and E Zorin.  Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory.  Int. Math. Res. Not. IMRN, (10):2885–2908, 2017.
  * [14] U. Betke, M. Henk, and J. M. Wills.  Successive-minima-type inequalities.  Discrete Comput. Geom., 9(2):165–175, 1993.
  * [15] H Blichfeldt.  Report on the theory of the geometry of numbers.  Bull. Amer. Math. Soc., 25(10):449–453, 1919.
  * [16] N Budarina.  Simultaneous Diophantine approximation in the real and $p$-adic fields with nonmonotonic error function.  Lith. Math. J., 51(4):461–471, 2011.
  * [17] Y Bugeaud, N Budarina, D Dickinson, and H O’Donnell.  On simultaneous rational approximation to a $p$-adic number and its integral powers.  Proc. Edinb. Math. Soc. (2), 54(3):599–612, 2011.
  * [18] JWS Cassels.  An introduction to the geometry of numbers.  Springer Science & Business Media, 2012.
  * [19] S Datta and A Ghosh.  Diophantine inheritance for p-adic measures.  https://arxiv.org/abs/1903.09362.
  * [20] B. M. M. de Weger.  Approximation lattices of $p$-adic numbers.  J. Number Theory, 24(1):70–88, 1986.
  * [21] K Falconer.  Fractal geometry.  John Wiley & Sons, Ltd., Chichester, third edition, 2014.  Mathematical foundations and applications.
  * [22] A Haynes.  The metric theory of $p$-adic approximation.  Int. Math. Res. Not. IMRN, 2010(1):18–52, 2010.
  * [23] M Henk.  Successive minima and lattice points.  Number 70, part I, pages 377–384. 2002.  IV International Conference in “Stochastic Geometry, Convex Bodies, Empirical Measures $\&$ Applications to Engineering Science”, Vol. I (Tropea, 2001).
  * [24] M Henk and F Xue.  On successive minima-type inequalities for the polar of a convex body.  Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113(3):2601–2616, 2019.
  * [25] J Huang.  The density of rational points near hypersurfaces.  Duke Math. J., 169(11):2045–2077, 2020.
  * [26] J Huang and J Liu.  Simultaneous Approximation on Affine Subspaces.  International Mathematics Research Notices, 11 2019.  rnz190.
  * [27] MN Huxley.  Area, lattice points, and exponential sums, volume 13 of London Mathematical Society Monographs. New Series.  The Clarendon Press, Oxford University Press, New York, 1996.  Oxford Science Publications.
  * [28] H Inoue, S Kamada, and K Naito.  Transference principle on simultaneous approximation problems of $p$-adic numbers and multidimensional $p$-adic approximation lattices.  Linear Nonlinear Anal., 3(2):239–249, 2017.
  * [29] H Inoue and K Naito.  The shortest vector problems in $p$-adic lattices and simultaneous approximation problems of $p$-adic numbers.  Linear Nonlinear Anal., 3(2):213–224, 2017.
  * [30] V Jarník.  Sur les approximations diophantiques des nombres $p$-adiques.  Rev. Ci. (Lima), 47:489–505, 1945.
  * [31] D Kleinbock and G Tomanov.  Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation.  Comment. Math. Helv., 82(3):519–581, 2007.
  * [32] E Lutz.  Sur les approximations diophantiennes linéaires $P$-adiques.  Actualités Sci. Ind., no. 1224. Hermann & Cie, Paris, 1955.
  * [33] A Mohammadi and A Salehi Golsefidy.  Simultaneous Diophantine approximation in non-degenerate $p$-adic manifolds.  Israel J. Math., 188:231–258, 2012.
  * [34] A Oliveira.  Khintchine’s theorem with rationals coming from neighborhoods in different places.  https://arxiv.org/abs/2006.14764.
  * [35] F Ramírez.  Khintchine types of translated coordinate hyperplanes.  Acta Arith., 170(3):243–273, 2015.
  * [36] F Ramírez, D Simmons, and F Süess.  Rational approximation of affine coordinate subspaces of Euclidean space.  Acta Arith., 177(1):91–100, 2017.
  * [37] J Schleischitz.  On the spectrum of Diophantine approximation constants.  Mathematika, 62(1):79–100, 2016.
  * [38] J Schleischitz.  Diophantine approximation on polynomial curves.  Math. Proc. Cambridge Philos. Soc., 163(3):533–546, 2017.
  * [39] V Sprindžuk.  Mahler’s problem in metric number theory.  Translated from the Russian by B. Volkmann. Translations of Mathematical Monographs, Vol. 25. American Mathematical Society, Providence, R.I., 1969.
  * [40] R. C. Vaughan and S. Velani.  Diophantine approximation on planar curves: the convergence theory.  Invent. Math., 166(1):103–124, 2006.
  * [41] B Wang and J Wu.  Mass transference principle from rectangles to rectangles in Diophantine approximation.  https://arxiv.org/abs/1909.00924.


# INTERSPEECH 2022 Audio Deep Packet Loss Concealment Challenge

###### Abstract

Audio Packet Loss Concealment (PLC) is the hiding of gaps in audio streams
caused by data transmission failures in packet switched networks. This is a
common problem, and of increasing importance as end-to-end VoIP telephony and
teleconference systems become the default and ever more widely used form of
communication in business as well as in personal usage.

This paper presents the INTERSPEECH 2022 Audio Deep Packet Loss Concealment
challenge. We first give an overview of the PLC problem, and introduce some
classical approaches to PLC as well as recent work. We then present the open
source dataset released as part of this challenge as well as the evaluation
methods and metrics used to determine the winner. We also briefly introduce
PLCMOS, a novel data-driven metric that can be used to quickly evaluate the
performance PLC systems. Finally, we present the results of the INTERSPEECH
2022 Audio Deep PLC Challenge, and provide a summary of important takeaways.

Index Terms: packet loss concealment, speech enhancement, real-time processing

##  1 Introduction

With the transition of digital voice communication away from circuit switched
analog telephony to end-to-end digital packet-switched telephone systems –
voice-over-IP (VoIP) telephony – the issue of how to best process and transmit
speech over packet networks has become increasingly important. One important
aspect of this is how to handle situations where packet transmission breaks
down because packets are lost entirely (packet loss) or arrive too late to be
used in a real time communication setting (loss due to excessive jitter).
Ideally, we would like to build systems that are robust to a small number of
lost packets, and degrade gracefully for longer loss bursts. This process of
restoring parts of the signal that are missing in a way that is minimally
disturbing to the users of a system is called Packet Loss Concealment (PLC).

Classically, PLC is treated as part of the design of the compression / coding
and decompression / decoding algorithms (codecs) used to transmit speech,
operating in the coding algorithms feature space. While these techniques allow
for basic smoothing over of losses, they can sometimes struggle even with
short losses: A study by Sun et al. [1] found that for codecs used in mobile
telephony, any packet loss during a voiced speech segment causes almost the
entirety of that segment to be degraded. Techniques that can be used
independently of the base PLC approach, such as forward error correction –
transmitting parts of frames multiple times in anticipation of a loss – can
help for short losses, but put additional strain on a network connection that
is already straining to transmit all packets. Lost packets manifest as
dropouts or audible distortion to the user – indeed, “distorted audio” is one
of the top problem tokens in Microsoft Teams post-call quality surveys.

A less studied but very promising approach that, due to advances in hardware
and algorithms, is performing PLC via machine learning and neural networks –
Deep PLC. By leveraging audio and call data to build models that can predict
future frames in a data-driven manner and without the strong assumptions
classical techniques require, it may be possible to transparently hide shorter
losses almost completely, and smooth over longer loss bursts in a less
distracting way than current PLC implementations are able to.

An issue that complicates current Deep PLC research is that as it stands,
there are no standard benchmark datasets and evaluation methods, so comparing
approaches is extremely limited. With the INTERSPEECH 2022 Deep PLC challenge,
we hope to rectify this situation by introducing a realistic evaluation
dataset that draws packet loss patterns from actual calls, a methodology for
evaluation that will facilitate the comparison of different approaches, and a
new objective metric that we hope will help researchers to iterate on their
approaches.

In the following paper, we will review classical PLC implementations, contrast
them with some recent work in Deep PLC, and finally, present the dataset,
evaluation methods and results from the INTERSPEECH 2022 Deep PLC Challenge.

##  2 Background and related work

###  2.1 Classical approaches

Classically, PLC is performed in the feature space of the codec used to
packetize, encode, and decode speech data, preceding the decoding step. This
is, of course, especially necessary if the codec in question relies on
information from consecutive packages in decoding. The techniques used for
this have been iteratively improved with newer generations of codecs, but have
in essence remained the same from the original work done for the Global System
for Mobile Telecommunication [2] to the modernized Adaptive Multi-Rate
Wideband codec used in UMTS [3] and the EVS codec used in Voice-over-LTE [4].

The basic principle of these approaches is to continue decoding as if the
change in the coded speech parameters from the last known good frames
continues according to some expert-crafted fixed prediction function
(linearly, in the simplest case), with gradual attenuation and replacement of
the signal with comfort noise. More modern codecs improve upon this scheme by
classifying missing frames into different types (e.g., silence, voiced speech,
non-periodic, …) and using different prediction schemes for different frame
types.

An additional technique employed in many codecs on top of these methods is
forward error correction: When bad network conditions are detected, the sender
can transmit redundant information about past frames so that short losses can
be better compensated for if the next frame after the loss is already
available. The downside of this technique is that it introduces network
overhead and additional latency. For a deeper survey on past proposed schemes,
refer to Thirunavukkarasu et al. [5].

###  2.2 Machine learning approaches

####  2.2.1 Audio inpainting

The field of _audio inpainting_ concerns itself with filling gaps in an audio
signal, with recent approaches largely favoring statistical methods. This is
related to the PLC problem, though in a relaxed form: It is often assumed that
the entire signal, other than the parts that need to be in-painted, is
available, and computation time is not generally considered.

A simple statistical approach is presented by Rodbro et al. [6]. They propose
to use a Hidden Markov Model on the pitch, gain, and spectral envelope of
packets that can then be used to either directly predict future frames,
directly fill gaps between frames, or be used as a frame type predictor to
choose a prediction scheme.

Bahat et al. [7] present a dictionary based scheme: They learn a dictionary of
audio blocks and continue an audio sequence by finding the best matching audio
block to continue the sequence with. To find the best match, they use both a
Markov model as well as the feature space distance between the start of the
block and the last known good part of the audio sequence. The dictionary is
created on the fly from correctly transmitted audio. This is a simple and
effective way to leverage audio data to fill audio gaps, though
computationally expensive and not easily adaptable beyond a single speaker.

Neural network approaches that consider the whole spectrogram are presented by
Kegler et al. [8] and Nair et al. [9]. The former treats the task of audio
inpainting as a vision task: They train a U-Net [10] model to map masked
spectrograms (magnitude and phase angle), optionally with an additional
channel indicating whether a part of the spectrum was masked or not, to the
unmasked spectrum, using a perceptual VGG network-based loss. They report an
improvement over a baseline linear predictive coding approach for most cases.
The latter use a joint time-frequency approach that also performs general
speech enhancement: They use a time-domain U-Net to fill gaps and then use a
frequency domain U-Net on the magnitude output of the first network to remove
distortions (while keeping the phase from the time-domain U-Net signal).

Others approach the problem from a multi-modal perspective: Both Zhou et al.
[11] and Morrone et al. [12] present approaches where an available video feed
of the speaker is used to assist in the restoration of lost audio segments,
using a convolutional neural network with adversarial loss and a recurrent
neural network approach respectively. While approaches such as these are
outside the purview of current deep PLC research, they may become relevant in
the context of video telephony, where in future techniques, available video
frames may help reconstruct the audio stream and vice-versa.

####  2.2.2 Real-time deep packet loss concealment

In recent years, both hardware and algorithms have advanced dramatically,
allowing neural network-based approaches to enter practical use in many areas
of speech processing. This is also happening for PLC. Here, we give an
overview of recent work targeting real-time, causal usage.

Stimbert et al.present an approach based on WaveRNN [13]: They condition a
WaveRNN on the recent time domain history as well as, through a convolutional
conditioning network operating in the frequency domain, more long term
history. Unlike other methods presented in this section, this network outputs
samples autoregressively, one by one, instead of outputting full blocks of
audio data at a time. They present methods for how their network can be used
for real-time inference, operating on the jitter buffer of the NetEQ codec
used in the Google Duo telecommunication software [14], and state that their
implementation of this method significantly improved user satisfaction
metrics.

One issue when training neural networks for audio generation is what loss to
use. Generative adversarial networks provide a framework within which a loss
does not need to be defined, but can be co-learned by training a discriminator
network that tries to classify a sample as either being generated or sampled
from the training set, and then training the generation network to try to fool
the discriminator. GAN-based approaches have been shown to be able to
efficiently generate audio waveforms [15]. A basic initial PLC approach based
on GANs is presented by Shi et al. [16]. They train a convolutional encoder-
decoder network operating on time domain audio blocks. They report comparable
quality in terms of several objective metrics (PESQ [17], STOI [18], SNR)
compared to a frequency-domain deep neural network even when this baseline has
perfect phase information available and only needs to predict magnitude.
Pascual et al. [19] present a GAN-based approach where the generator input is
the Mel-spectrogram of the available signal, and the output is the time-domain
continuation of this signal. They show that this method improves upon several
baselines (including some of the systems used in real codecs, such as NetEQ
and ITU G722.1) in terms of Mel-Cepstral Distortion [20] as well as SESQUA
[21]. Finally, Wang et al. [22] present a GAN-based system with a fully time-
domain U-Net style convolutional generator and mixed time/frequency domain
discriminator, which allows their adversarial loss to both capture fine short-
term details in the waveform as well as long-term relationships in the
spectrum. They evaluate in terms of PESQ, STOI, and SNR and are able to show
an improvement over several strong baselines – even over non-causal models
when their own model is being used causally.

Other methods show that Deep PLC systems can be trained even with simpler
losses. Lin et al. [23] present a convolutional-recurrent model performing
next frame prediction in the time domain, trained to minimize the mean
absolute error, with or without look-ahead. In addition to the usual metrics,
they also measure speech recognizer word error rate, and are able to show that
Deep PLC has the potential to improve this metric as well. Similarly, Mohamed
et al. [24] present a recurrent neural network for packet loss concealment in
the context of far-end emotion recognition. They show that using such a
network before their emotion classifier improves its accuracy, further
illustrating the potential of Deep PLC to improve the performance of
downstream speech processing tasks.

##  3 Challenge description

The objective of the INTERSPEECH 2022 Audio Deep Packet Loss Concealment
challenge is to fill regions where audio has been lost due to packet loss in
such a way as to ideally hide these losses from listeners and to maximize
intelligibility.

Note that effects from de- and encoding or buffering and time stretching are
not considered as part of this challenge.

###  3.1 Dataset and evaluation

The dataset provided for the INTERSPEECH 2022 Audio Deep Packet Loss
Concealment challenge consists of a set of audio and meta-data files, divided
into three splits: A train split, validation split and blind test split. The
training and validation splits contain, for each included audio sample:

  * •

A clean audio file

  * •

A “lossy” audio file with packet loss regions zeroed out

  * •

A text file with loss metadata

The metadata indicates, with one line for each 20 millisecond segment of
audio, whether the packet has been lost (line contains a “1”) or not (line
contains a “0”). The “lossy” audio files have the corresponding segments
zeroed out. Additionally, the dataset includes, for each split, a file with
file-level metadata about the included clips in csv format. The training set
is intended as a quick-start set for training PLC models, though participants
were free to use any other datasets to improve their models. The blind set
release did not include clean audio.

The training split includes a total of 23184 clips, while the validation and
blind test split each include 966 clips.

Training and validation splits were released to participants on January 19th,
2022. The blind split data was released on March 1st, 2022, after which
participants had one week to generate and submit results. Blind split
reference data was released after the conclusion of the challenge. The dataset
is now available as a fully open source dataset for use in further evaluations
by any interested researcher111https://github.com/microsoft/PLC-Challenge.

####  3.1.1 Dataset construction

The dataset was constructed by a stratified sampling of actual packet loss
traces observed in calls made by Microsoft Teams users, applied to randomly
chosen segments of audio from a podcast dataset.

Figure 1: CMOS for lossy audio files for different maximum burst sizes and
loss percentage quantiles included in our dataset.

The traces were sampled as follows: First, 10 second segments with at least
one lost packet were randomly extracted from packet loss traces from Teams
calls. These segments were then divided into three subsets according to the
maximum burst loss length in the trace:

  * •

up to 120 milliseconds (504 blind set clips)

  * •

between 120 and 320 milliseconds (308 blind set clips)

  * •

between 320 and 1000 milliseconds (154 blind set clips)

Traces with burst losses longer than 1000 milliseconds were discarded. Each
subset was then divided into 14 cells according to packet percentage loss
quantiles. Finally, an equal amount of traces was sampled from each cell (with
more traces being sampled for the subsets with shorter maximum burst losses).

To create audio with packet losses, clips of audio of approximately 10 seconds
of length were sampled from a base public domain podcast dataset (LibriVox
Community Podcasts), cutting in low volume regions to not split up words. The
losses from the sampled traces were then applied to the resulting audio files
by zeroing out the corresponding regions in the audio clips.

Figure 1 shows CMOS scores (compare section 3.1.3 for the different burst
subsets and packet loss fraction percentiles in the blind set. For low burst
lengths, a smooth, relatively linear relationship between loss and CMOS is
visible. For the longer burst lengths, the gradient is shifted substantially
towards worse scores (with a lot of noise for the very high burst lengths
likely owing to the relatively small amount of clips per cell).

####  3.1.2 Latency and compute restrictions

In practice, PLC is used as part of an audio telecommunications system. This
means that Deep PLC models have tight latency requirements. In the 2022
INTERSPEECH Deep PLC Challenge, the requirements given for models were as
follows:

  * •

The PLC must take less than the stride time to process a frame on an Intel
Core i5 quad-core machine clocked at 2.4 GHz or equivalent processors.

  * •

Total algorithmic latency including the frame size, stride time, and any look-
ahead must be $\leq$ 20ms.

####  3.1.3 Evaluation

The evaluation criterion is split into two parts, weighted equally: Crowd-
Sourced _Comparative Mean Opinion Score_ (CMOS), and _Word Accuracy_ (WAcc).
The Mean Opinion Score is calculated by having raters rate clips using a
crowd-sourcing framework [25], aiming for five ratings for each clip in the
blind set. Since in PLC, it is possible to have clips where a system may fill
a long gap with silence where in the original, speech was present, raters are
given the reference for every file, and asked to rate the given file compared
to this reference on a 7 point categorical scale ranging from -3 to 3 – this
is called comparative category rating. For details on evaluation methodology,
please refer to our previous work. [25]

The word accuracy is calculated using Microsoft Azure Cognitive Services
speech recognition by subtracting the returned _Word Error Rate_ (WER) from 1.
The score ranges from 0 to 1, where 1 is a perfect score.

To calculate the final score, we normalize the CMOS range from 0 (the same as
the reference) to -3 (much worse than the reference) to a range of 0 to 1 such
that higher is better, and take the mean of the normalized CMOS and the word
accuracy. The final score by which we rank submissions is calculated as:

$\text{score}=\frac{\left(1+\frac{\text{CMOS}}{3}\right)+\left(1-\text{WER}\right)}{2}$

###  3.2 PLCMOS

Table 1: 2022 INTERSPEECH Audio Deep PLC Challenge Results. Teams marked as “tied”: Difference not significant ($p\geq 0.05$). Parameter count and structure given where known (TD: Time-Domain, FD: Frequency Domain). Rank | Team ID | PLCMOS | DNSMOS | CMOS | WAcc | Final score | #Parameters | Approach
---|---|---|---|---|---|---|---|---
| Clean (reference) | 4.51 | 3.89 | 0 | 0.97 | 0.98 | - | -
1 | #12 | 4.28 | 3.80 | -0.55 | 0.88 | 0.85 |  ~2.36M | TD U-Net
2 | #1 | 3.74 | 3.79 | -0.64 | 0.88 | 0.84 |  ~5.9M | RNN + LPCNet
3 (tied) | #9 | 3.83 | 3.68 | -0.81 | 0.87 | 0.80 |  | TD Encoder-Decoder
3 (tied) | #14 | 3.98 | 3.69 | -0.84 | 0.86 | 0.79 |  ~4.97M | TD Sequence-to-One
5 | #11 | 3.28 | 3.51 | -1.10 | 0.86 | 0.75 |  ~36k | FD Conv. Net
6 | #10 | 3.48 | 3.74 | -1.04 | 0.83 | 0.74 |  |
| Zero-filling baseline | 2.90 | 3.44 | -1.23 | 0.86 | 0.73 | - | -
7 | #6 | 2.90 | 3.48 | -1.31 | 0.86 | 0.71 |  ~3.77M |
Table 2: 2022 INTERSPEECH Audio Deep PLC Challenge: Metrics for non-eligible models. Team ID | PLCMOS | DNSMOS | CMOS | WAcc.
---|---|---|---|---
#2 | 4.406 | 3.958 | -0.279 | 0.854
#13 | 3.274 | 3.453 | -1.115 | 0.853
#3 | 3.371 | 3.293 | -1.284 | 0.852

To help participants evaluate their systems, we developed and provided access
to PLCMOS222https://github.com/microsoft/PLC-Challenge/tree/main/PLCMOS, a
neural network model trained to estimate the score a human rater would assign
to a given audio file. There are multiple models provided: A semi-intrusive
model that requires a reference, but does not require the reference to be
aligned with the lossy signal, and a non-intrusive model that functions
entirely without reference audio. The model employs a convolutional-recurrent
encoder structure and is trained on ground truth labels produced by P.808 ACR
labelling [25]. It produces a score between 1 and 5 as an estimate of human
mean opinion scores. We report PLCMOS scores for our participants in Table 1.
Overall, on the data evaluated for this challenge, PLCMOS and subjective CMOS
showed a Pearson correlation coefficient of ~0.95 and a Spearman rank
correlation coefficient of ~0.93, compared to a PCC of ~0.89 and SRCC of ~0.90
for DNSMOS [26], which hasn’t been trained on packet loss data. We plan to
further improve this system in the future with more data, including the data
gathered during this challenge.

##  4 Results

We present the results of our evaluation in Table 1, including PLCMOS and
DNSMOS [26] metrics. In addition to participant results, we also list the
metrics for a zero-filling baseline, which simply replaced the losses with
silence – the outputs of this baseline comprises the “lossy” dataset made
available to the contestants.

CMOS results were obtained using the Amazon Mechanical Turk service and [25],
targeting 5 ratings per clip. A total of 421 raters participated in the
evaluation. An initial run had a rejected answer rate of ~30% due to failed
trapping clips or raters submitting more than the allowed amount of ratings.
Rejected ratings were re-ran. After removing raters with low acceptance rates
($<$90%) and with few completed rating tasks ($<$3), the final results
presented here are based on at least 4 votes per clip for 75% of clips
submitted for rating, with a confidence interval of ~0.035 for the CMOS score.

Word accuracy was calculated based on 944 clips of the blind set. 22 clips
containing multi-lingual or non-speech audio were excluded from WER evaluation
because the speech recognition service used could not reliably produce
transcriptions of these clips. Ground truth transcriptions were created using
Azure Cognitive Services to transcribe the clean audio files, followed by
human inspection and correction where needed.

We performed statistical tests to get a better understanding of how
significant the differences between models are. After establishing that
significant differences exist using ANOVA testing, we performed post-hoc
pairwise t-tests for the CMOS votes, using the same procedure as in our
previous challenges. We find that, while most differences are significant at a
level of $p<0.05$, we cannot establish significant differences between the
Alibaba inc. and Oldenburg University teams. In our final results, we denote
these teams as tied.

In addition to the models presented in Table 1, some models submitted for the
challenge used a larger algorithmic latency than intended due to oversights in
modeling. The prevalence of this issue underscores the need to carefully
verify constraints when building models for use in real-time and low-latency
applications. The metrics for these submissions are listed in Table 2 – while
we could not consider them for this challenge, they are nevertheless
interesting, especially in regard to potential future challenges going towards
a full neural jitter buffer, where a larger look-ahead will in some cases be
appropriate.

##  5 Conclusion

This paper has presented the results of the INTERSPEECH 2022 Audio Deep PLC
Challenge. Looking back, we believe this challenge has managed to achieve its
intended goal – to stimulate research into neural network-based PLC and to
bring researchers interested in this topic together. Some key takeaways
include:

  * •

While imperfect, objective metrics are still quite predictive of the final
results. We’ll continue improving PLCMOS to facilitate the development of
ever-better PLC systems.

  * •

Scores in the challenge were quite close, especially towards the top end, and
significant differences could not always be established. For future
challenges, we will look into constructing a more difficult task, possibly
going into the direction of a full neural jitter buffer so that models can
infill and modify the time scale with which samples are played back, within
some given maximum latency.

  * •

One effect we did not expect is the relative prevalence of accidental leaks of
future information beyond the allowed maximum algorithmic latency. For future
challenges, we will try to equip participants with tools they can use to
easily verify that their model fits the constraints provided.

We would like to thank all participants for their efforts in creating and
submitting models, and INTERSPEECH for hosting the Audio Deep Packet Loss
Concealment Challenge.

## References

  * [1] L. F. Sun, G. Wade, B. M. Lines, and E. C. Ifeachor, “Impact of Packet Loss Location on Perceived Speech Quality,” in _In 2nd IP-Telephony Workshop_ , 2001, pp. 114–122.
  * [2] K. Hellwig, P. Vary, D. Massaloux, J. Petit, C. Galand, and M. Rosso, “Speech codec for the European mobile radio system,” in _1989 IEEE Global Telecommunications Conference and Exhibition ’Communications Technology for the 1990s and Beyond’_ , Nov. 1989, pp. 1065–1069 vol.2.
  * [3] 3rd Generation Partnership Project, “Adaptive Multi-Rate (AMR) speech codec; Error concealment of lost frames,” in _Adaptive Multi-Rate (AMR) speech codec_ , 2004.
  * [4] J. Lecomte, T. Vaillancourt, S. Bruhn, H. Sung, K. Peng, K. Kikuiri, B. Wang, S. Subasingha, and J. Faure, “Packet-loss concealment technology advances in EVS,” in _2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , Apr. 2015, pp. 5708–5712, iSSN: 2379-190X.
  * [5] E. Thirunavukkarasu and E. Karthikeyan, “A survey on VoIP packet loss techniques,” _International Journal of Communication Networks and Distributed Systems_ , vol. 14, no. 1, pp. 106–116, Jan. 2015, publisher: Inderscience Publishers.
  * [6] C. Rodbro, M. Murthi, S. Andersen, and S. Jensen, “Hidden Markov model-based packet loss concealment for voice over IP,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 14, no. 5, pp. 1609–1623, Sep. 2006, conference Name: IEEE Transactions on Audio, Speech, and Language Processing.
  * [7] Y. Bahat, Y. Y. Schechner, and M. Elad, “Self-content-based audio inpainting,” _Signal Processing_ , vol. 111, pp. 61–72, Jun. 2015\.
  * [8] M. Kegler, P. Beckmann, and M. Cernak, “Deep Speech Inpainting of Time-Frequency Masks,” in _Interspeech 2020_. ISCA, Oct. 2020, pp. 3276–3280.
  * [9] A. A. Nair and K. Koishida, “Cascaded Time + Time-Frequency Unet For Speech Enhancement: Jointly Addressing Clipping, Codec Distortions, And Gaps,” in _ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , Jun. 2021, pp. 7153–7157, iSSN: 2379-190X.
  * [10] O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional Networks for Biomedical Image Segmentation,” _arXiv:1505.04597 [cs]_ , May 2015, arXiv: 1505.04597 version: 1.
  * [11] H. Zhou, Z. Liu, X. Xu, P. Luo, and X. Wang, “Vision-Infused Deep Audio Inpainting,” in _2019 IEEE/CVF International Conference on Computer Vision (ICCV)_. Seoul, Korea (South): IEEE, Oct. 2019, pp. 283–292.
  * [12] G. Morrone, D. Michelsanti, Z.-H. Tan, and J. Jensen, “Audio-Visual Speech Inpainting with Deep Learning,” in _ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , Jun. 2021, pp. 6653–6657, iSSN: 2379-190X.
  * [13] N. Kalchbrenner, E. Elsen, K. Simonyan, S. Noury, N. Casagrande, E. Lockhart, F. Stimberg, A. v. d. Oord, S. Dieleman, and K. Kavukcuoglu, “Efficient Neural Audio Synthesis,” _arXiv:1802.08435 [cs, eess]_ , Feb. 2018, arXiv: 1802.08435 version: 1.
  * [14] “Improving Audio Quality in Duo with WaveNetEQ,” http://ai.googleblog.com/2020/04/improving-audio-quality-in-duo-with.html. Archived: https://web.archive.org/web/20220309043235/http://ai.googleblog.com/2020/04/improving-audio-quality-in-duo-with.html.
  * [15] J. Kong, J. Kim, and J. Bae, “HiFi-GAN: Generative Adversarial Networks for Efficient and High Fidelity Speech Synthesis,” _arXiv:2010.05646 [cs, eess]_ , Oct. 2020, arXiv: 2010.05646.
  * [16] Y. Shi, N. Zheng, Y. Kang, and W. Rong, “Speech Loss Compensation by Generative Adversarial Networks,” in _2019 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC)_ , Nov. 2019, pp. 347–351, iSSN: 2640-0103.
  * [17] A. Rix, J. Beerends, M. Hollier, and A. Hekstra, “Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs,” in _2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221)_ , vol. 2. Salt Lake City, UT, USA: IEEE, 2001, pp. 749–752.
  * [18] C. H. Taal, R. C. Hendriks, R. Heusdens, and J. Jensen, “An Algorithm for Intelligibility Prediction of Time–Frequency Weighted Noisy Speech,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 19, no. 7, pp. 2125–2136, Sep. 2011, conference Name: IEEE Transactions on Audio, Speech, and Language Processing.
  * [19] S. Pascual, J. Serrà, and J. Pons, “Adversarial Auto-Encoding for Packet Loss Concealment,” _arXiv:2107.03100 [cs, eess]_ , Jul. 2021, arXiv: 2107.03100.
  * [20] R. Kubichek, “Mel-cepstral distance measure for objective speech quality assessment,” in _Proceedings of IEEE Pacific Rim Conference on Communications Computers and Signal Processing_ , vol. 1, May 1993, pp. 125–128 vol.1.
  * [21] J. Serrà, J. Pons, and S. Pascual, “SESQA: semi-supervised learning for speech quality assessment,” _arXiv:2010.00368 [cs, eess]_ , Feb. 2021, arXiv: 2010.00368.
  * [22] J. Wang, Y. Guan, C. Zheng, R. Peng, and X. Li, “A temporal-spectral generative adversarial network based end-to-end packet loss concealment for wideband speech transmission,” _The Journal of the Acoustical Society of America_ , vol. 150, no. 4, pp. 2577–2588, Oct. 2021, publisher: Acoustical Society of America.
  * [23] J. Lin, Y. Wang, K. Kalgaonkar, G. Keren, D. Zhang, and C. Fuegen, “A Time-Domain Convolutional Recurrent Network for Packet Loss Concealment,” in _ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , Jun. 2021, pp. 7148–7152, iSSN: 2379-190X.
  * [24] M. M. Mohamed and B. W. Schuller, “ConcealNet: An End-to-end Neural Network for Packet Loss Concealment in Deep Speech Emotion Recognition,” _arXiv:2005.07777 [cs, eess]_ , May 2020, arXiv: 2005.07777.
  * [25] B. Naderi and R. Cutler, “An Open Source Implementation of ITU-T Recommendation P.808 with Validation,” _INTERSPEECH_ , pp. 2862–2866, Oct. 2020, arXiv: 2005.08138.
  * [26] C. K. A. Reddy, V. Gopal, and R. Cutler, “DNSMOS: A Non-Intrusive Perceptual Objective Speech Quality metric to evaluate Noise Suppressors,” _arXiv:2010.15258 [cs, eess]_ , Feb. 2021, arXiv: 2010.15258.