1 Introduction
The theory of metric Diophantine approximation on manifolds began with Mahler’s conjecture which states that almost every point on the Veronese curve
$\mathcal V_n := \{(x,x^2,\dots ,x^n):x \in \mathbb R\}$
is not very well approximable. Sprindžuk [Reference Sprindzuk20] proved this conjecture in
$1960$
and conjectured [Reference Sprindzuk21] that the conclusion of Mahler’s conjecture is true for any nondegenerate analytic submanifold of
$\mathbb R^n$
. This was strengthened by Baker and is commonly referred to as the Baker–Sprindžuk conjecture. After some partial results, the conjecture in its full generality was finally resolved in 1998 by Kleinbock and Margulis [Reference Kleinbock and Margulis15] using techniques from homogeneous dynamics. Beresnevich and Velani and others [Reference Badziahin, Beresnevich and Velani1–Reference Beresnevich and Velani4] observed that, sometimes, the results of homogeneous Diophantine approximation can be transferred to the inhomogeneous context by a transference principle. In [Reference Beresnevich and Velani4], Beresnevich and Velani proved many results of inhomogeneous Diophantine approximation, including the inhomogeneous Baker–Sprindžuk conjecture in this way. Our aim is to explore this theme in the context of function fields.
Consider the function field
$\mathbb {F}_{q}(X)$
, where
$q= p^d$
, p is a prime and
$d\in \mathbb {N}$
. Define a nonarchimedean absolute value
$|\cdot |$
on
$\mathbb {F}_{q}(X)$
by
$$ \begin{align*} |0| := 0\quad \text{ and} \quad \bigg|\frac{f}{g}\bigg| =e^{{\operatorname{deg}}\,f - {\operatorname{deg}}\,g} \quad\text{for } f, g\in \mathbb{F}_{q}[X] \setminus \{0 \}\,.\end{align*} $$
By
$\mathbb {F}_{q}((X^{-1})),$
we denote the field of Laurent series in
$X^{-1}$
over the finite field
$\mathbb {F}_q$
. Note that
$\mathbb {F}_{q}((X^{-1}))$
is the completion of
$\mathbb {F}_{q}(X)$
with respect to the absolute value
$|\cdot |$
and the absolute value of
$\mathbb {F}_{q}(X)$
extends to an absolute value of
$\mathbb {F}_{q}((X^{-1}))$
as follows. For
$f \in \mathbb {F}_{q}((X^{-1})) \setminus \{0\}$
, we can write
$$ \begin{align*}f=\displaystyle \sum_{\ell\leq \ell_{0}} {\alpha}_{\ell} X^{\ell} \quad\text{where }\ell_0 \in \mathbb{Z},\,{\alpha}_{\ell}\in \mathbb{F}_{q} \text{ and } {\alpha}_{\ell_0} \neq 0.\end{align*} $$
We define
$|f|:=e^{\ell _0}$
. With respect to this absolute value,
$\mathbb {F}_{q}((X^{-1}))$
is an ultrametric, complete and separable metric space. Every local field of positive characteristic is isomorphic to some
$\mathbb {F}_{q}((X^{-1}))$
.
From now on,
$\mathbb {F}_{q}[X]$
and
$\mathbb {F}_{q}((X^{-1}))$
will be denoted by
$\Lambda $
and F, respectively. We equip
$F^n$
with the supremum norm:
$$ \begin{align*}||\mathbf{y}|| := \displaystyle \max_{1\leq i\leq n} |y_i| \quad\text{for } \mathbf{y}=(y_1,y_2,\ldots,y_n)\in F^{n}.\end{align*} $$
In the topology induced by the sup norm,
$\Lambda ^n$
is discrete in
$F^n$
and the local compactness of F implies that
$F^n$
is locally compact. By
$\lambda $
, we denote the Haar measure on
$F^n$
such that
$\lambda (\{\mathbf {x} \in F^n : \|\mathbf {x}\| \leq 1 \})=1$
.
Diophantine approximation over function fields has been studied extensively since Mahler developed the geometry of numbers in this context [Reference Mahler18]. We refer to the survey in [Reference Lasjaunias17] and to [Reference Das and Ganguly7, Reference Kim and Nakada13, Reference Robert, Hussain, Shulga and Ward19] for recent developments. Let us first recall Dirichlet’s theorem over function fields.
Theorem 1.1 [Reference Ganguly and Ghosh9, Theorem 2.1].
Let
$m,n \in \mathbb N, \ell =m+n$
and
$$ \begin{align*} \alpha^+ := \bigg\{ \mathbf{t}:=(t_1,\dots,t_l) \in \mathbb{Z}_{+}^{\ell}: \sum_{i=1}^{m}t_i=\sum_{i=1}^{n}t_{m+i}\bigg\}. \end{align*} $$
Let Z be an
$m \times n $
matrix over
$F.$
Then for any
$\mathbf {t} \in \alpha ^+$
, there exist solutions
${\mathbf {q}=(q_1,\dots ,q_n) \in \Lambda ^n \setminus \{0\}}$
and
$\mathbf {p}=(p_1,\dots ,p_m) \in \Lambda ^m$
of the system of inequalities:
$$ \begin{align*} \left \{ \begin{array}{@{}rl} |Z_{i}\mathbf{q}+p_i|< e^{-t_{i}} &\text{for } i=1,2,\ldots,m, \\ |q_j|< e^{t_{m+j}} & \text{for } j=1,2,\ldots,n\,, \end{array} \right. \end{align*} $$
where
$Z_1, \dots ,Z_m$
are the row vectors of
$Z.$
Here,
$\mathbb Z_+$
denotes the set of nonnegative integers. We will not pursue this much generality, but only consider unweighted Diophantine approximation. We denote the set of all
$m \times n$
matrices over F by
$F^{m\times n}.$
Definition 1.2 [Reference Ganguly, Ghosh, Moree, Pohl, Snoha and Ward10].
Given
$Y \in F^{m \times n} $
and 
we define the inhomogeneous exponent 
of Y as the supremum of real numbers
$\omega \geq 0$
such that there exists a solution
$(\mathbf {p}, \mathbf {q}) \in \Lambda ^m \times (\Lambda ^n \setminus \{0\})$
to the system of inequalities
for arbitrarily large
$T \geq 1. $
Definition 1.3. Given
$Y \in F^{m \times n} $
and 
we define the inhomogeneous uniform exponent 
of Y as the supremum of real numbers
$\hat {\omega } \geq 0$
such that there exists a solution
$(\mathbf {p}, \mathbf {q}) \in \Lambda ^m \times (\Lambda ^n \setminus \{0\})$
to the system of inequalities
for all sufficiently large
$T \geq 1. $
Definition 1.4. Given
$Y \in F^{m \times n} $
and 
we define the multiplicative inhomogeneous exponent 
of Y as the supremum of real numbers
${\omega } \geq 0$
such that there exists a solution
$(\mathbf {p}, \mathbf {q}) \in \Lambda ^m \times (\Lambda ^n \setminus \{0\})$
to the system of inequalities
for arbitrarily large
$T \geq 1, $
where
$$ \begin{align*} \prod (\mathbf{y}):= \prod_{j=1}^{m} |y_j| \quad \text{and} \quad {\prod}_+ (\mathbf{q}):= \prod_{i=1}^{n} \max \{1,|q_i|\} \end{align*} $$
for
$\mathbf {y}=(y_1,\dots ,y_m) \in F^m$
and
$\mathbf q=(q_1,\dots ,q_n) \in \Lambda ^n.$
In a similar manner, given any
$Y \in F^{m \times n}$
and 
, we define the multiplicative inhomogeneous uniform exponent 
of Y. Note that the analogous homogeneous exponents are provided by the special case of 
where 
, that is, 
and 
. Also
By Dirichlet’s theorem,
$\omega (Y) \geq 1$
for all
$Y \in F^{m \times n}.$
Hence,
$\omega ^{\times }(Y) \geq 1.$
An easy application of the Borel–Cantelli lemma shows that
$\omega (Y)=1$
and
$\omega ^{\times }(Y)=1$
for
$\lambda $
-almost every
$Y \in F^{m \times n}.$
We say that
$Y \in F^{m \times n}$
is very well approximable (VWA) if
$\omega (Y)>1$
and
$Y \in F^{m \times n}$
is very well multiplicatively approximable (VWMA) if
$\omega ^{\times } (Y)>1.$
The function field analogue of the Baker–Sprindžuk conjecture states that almost every point on an analytic nondegenerate submanifold
$\mathcal M$
of
$F^n,$
identified with either columns
$F^{n \times 1}$
(in simultaneous Diophantine approximation) or rows
$F^{1 \times n}$
(in dual Diophantine approximation), is not VWMA with respect to the natural measure on
$\mathcal M.$
In the language of exponents, if
$\mathcal M$
is any analytic nondegenerate submanifold of
$F^n$
, then
$$ \begin{align} \omega^{\times}(Y)=1 \quad\text{for almost every } Y \in \mathcal M, \end{align} $$
and this clearly implies that
$$ \begin{align} \omega(Y)=1 \quad\text{for almost every } Y \in \mathcal M. \end{align} $$
This was settled by Ghosh in [Reference Ghosh11]. The manifolds of
$F^n$
which satisfy (1.2) and (1.3) are referred to as strongly extremal and extremal, respectively. In view of the above discussion, we have the following theorem.
Theorem 1.5 [Reference Ghosh11, Theorem 3.7].
Let
$\mathcal {M}$
be an analytic nondegenerate submanifold of
$F^n.$
Then
$\mathcal M$
is strongly extremal.
Kleinbock, Lindenstrauss and Weiss introduced the idea of measures being extremal rather than sets. Following their terminology, a measure
$\mu $
supported on a subset of
$F^{m \times n}$
is extremal (strongly extremal) if
$\omega (Y)=1$
(
$\omega ^{\times }(Y)=1$
) for
$\mu $
-almost every point
$Y \in F^{m \times n}$
. Due to Khintchine’s transference principle, we say that a measure
$\mu $
on
$F^n$
is (strongly) extremal if it is (strongly) extremal as a measure on
$F^{1 \times n}$
or
$F^{n \times 1}.$
Theorem 1.6. Let
$\mu $
be a friendly measure on
$F^n$
. Then
$\mu $
is strongly extremal.
We will define the notion of friendly measure in Section 3. Friendly measures provide us with a large class of measures on
$F^n$
that includes the natural measure on a nondegenerate analytic manifold. The proof of Theorem 1.6 is analogous to that of Theorem 11.1 and Corollary 11.2 of [Reference Kleinbock and Tomanov16]. Hence, Theorem 1.6 implies Theorem 1.5.
Definition 1.7. A measure
$\mu $
supported on a subset of
$F^{m \times n} $
is said to be inhomogeneously extremal if, for all 
, we have 
. We call
$\mu $
inhomogeneously strongly extremal if, for all 
, we have 
for
$\mu $
-almost every
$Y \in F^{m \times n}$
.
It is easy to see that this naturally generalises the notion of extremality in the homogeneous case (
). In the homogeneous case, strong extremality implies extremality but it is not at all clear why an inhomogeneously strong extremal measure would be inhomogeneously extremal. The following proposition, proved in Section 4, answers this question.
Proposition 1.8. Suppose that
$\mu $
be a measure supported on a subset of
$F^{m \times n}.$
Then
$$ \begin{align*} \mu \text{ is inhomogeneously strongly extremal}\! \implies\! \mu \text{ is inhomogeneously extremal}. \end{align*} $$
As mentioned earlier, in the homogeneous case, both the simultaneous and dual forms of Diophantine approximation lead to the same notion of extremality due to Khintchine’s transference principle. As there is no Khitchine’s transference principle in the inhomogeneous case, one has to deal with simultaneous and dual forms of Diophantine approximation separately.
Definition 1.9. A measure
$\mu $
supported on a subset of
$F^n$
is said to be dually inhomogeneously (strongly) extremal if
$\mu $
is inhomogeneously (strongly) extremal on
$F^{1 \times n}$
. We call
$\mu $
simultaneously inhomogeneously (strongly) extremal if
$\mu $
is inhomogeneously (strongly) extremal on
$F^{n \times 1}.$
Furthermore, we say that
$\mu $
is inhomogeneously (strongly) extremal if
$\mu $
is both dually and simultaneously inhomogeneously (strongly) extremal.
In [Reference Ganguly, Ghosh, Moree, Pohl, Snoha and Ward10], Ganguly and Ghosh proved the inhomogeneous Sprindžuk conjecture over a field of positive characteristic.
Theorem 1.10 [Reference Ganguly, Ghosh, Moree, Pohl, Snoha and Ward10, Theorem 1.1].
Let
$\mathcal {M}$
be an analytic nondegenerate submanifold of
$F^n.$
Then
$\mathcal M$
is inhomogeneously extremal.
2 Main results
In this section, we describe the main results of this paper.
Theorem 2.1.
(A) Let
$\mu $
be an almost everywhere contracting measure on
$F^{m \times n}$
. Then
$$ \begin{align*} \mu\text{ is extremal} \!\iff\! \mu\text{ is inhomogeneously extremal}. \end{align*} $$
(B) Let
$\mu $
be an almost everywhere strongly contracting measure on
$F^{m \times n}$
. Then
$$ \begin{align*} \mu\text{ is strongly extremal} \!\iff\! \mu\text{ is inhomogeneously strongly extremal}. \end{align*} $$
We will define a (strongly) contracting measure in the next section. Strongly contracting measures form a class of measures containing friendly measures. Hence, in view of Theorem 2.1 and Corollary 3.5, we have the following inhomogeneous version of Theorem 1.6.
Theorem 2.2. Let
$\mu $
be a friendly measure on
$F^n$
. Then
$\mu $
is inhomogeneously strongly extremal.
It is well known that the natural measure supported on an analytic nondegenerate manifold is friendly. Hence, as an immediate consequence of Theorems 2.1, 1.5 and Corollary 3.5, we get the inhomogeneous Baker–Sprindžuk conjecture.
Theorem 2.3. Let
$\mathcal {M}$
be an analytic nondegenerate submanifold of
$F^n.$
Then
$\mathcal M$
is inhomogeneously strongly extremal.
Furthermore, we prove the following upper bounds for the nonextremal case.
Theorem 2.4.
(I) Let
$\mu $
be a measure on
$F^{m \times n}$
which is contracting almost everywhere. Suppose that
$\omega (Y)=\eta $
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
Then, for all 
,
(II) Let
$\mu $
be a measure on
$F^{m \times n}$
which is strongly contracting almost everywhere. Suppose that
$\omega ^{\times }(Y)=\eta $
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
Then, for all 
,
The above theorem is the function field analogue of [Reference Ghosh and Marnat12, Theorem 2.2].
3 Strongly contracting and friendly measures
We retain the notation and terminologies of Beresnevich and Velani from [Reference Beresnevich and Velani4]. Let X be a metric space and
$B \subseteq X$
be a ball. For
$a>0, aB$
denotes the ball with the same centre as B and radius a times the radius of B. We say that a measure
$\mu $
on X is nonatomic if
$\mu (\{x\})=0$
for any
$x \in X.$
The support of a measure is defined to be the smallest closed set S such that
$\mu (X \setminus S)=0.$
We say that
$\mu $
is doubling if there exists a constant
$c>0$
such that for any ball B with centre in S, we have
$ \mu (2B) \leq c \mu (B)$
.
Consider the plane
$ \mathcal {L}_{\mathbf {b},\mathbf {c}} := \{Y \in F^{m \times n}: Y \mathbf {b} + \mathbf {c}=0\} $
for
$\mathbf {b} \in F^n$
with
$\|\mathbf {b}\|=1$
and
${\mathbf {c} \in F^m}$
. For 
, we define the 
-neighbourhood of the plane
$\mathcal {L}_{\mathbf {b},\mathbf {c}} $
by
where
$Y_i$
is the ith row of
$Y.$
If
$\varepsilon _1=\cdots =\varepsilon _m=\varepsilon , $
we simply denote it by
$ \mathcal {L}_{\mathbf {b},\mathbf {c}}^{(\varepsilon )}.$
Definition 3.1. Let
$\mu $
be a finite, nonatomic and doubling Borel measure on
$F^{m \times n}.$
Then
$\mu $
is said to be strongly contracting if there exists
$C,\alpha ,r_0>0$
such that for any plane
$\mathcal {L}_{\mathbf {b},\mathbf {c}},$
any 
with
$\min \{\varepsilon _j : 1 \leq j \leq m\} < r_0$
and
$0<\delta <1,$
the following property is satisfied: for all 
, there is an open ball B centred at Y such that
and
We call the measure
$\mu $
contracting if the same property holds with
${\varepsilon _1 = \cdots = \varepsilon _m =\varepsilon }$
. Further,
$\mu $
is (strongly) contracting almost everywhere if for
$\mu $
-almost every point
$Y_0 \in F^{m \times n}$
, there is a neighbourhood V of
$Y_0$
such that the restriction
$\mu |_V$
of
$\mu $
to V is (strongly) contracting.
First, note that if a measure
$\mu $
is strongly contracting then it is contracting. Next, we recall the notion of friendly measures from [Reference Kleinbock, Lindenstrauss and Weiss14, Reference Kleinbock and Tomanov16]. For this, we first need to define nonplanarity and
$(C,\alpha )$
-decaying for a measure
$\mu .$
Let
$\mu $
be a Borel measure on
$F^n$
and
$S:= \text {the support of } \mu .$
We say
$\mu $
is nonplanar if
$\mu (\mathcal L )=0$
for any hyperplane
$\mathcal L$
. Given a hyperplane
$\mathcal L$
and a ball B of
$F^n$
with
$\mu (B)>0,$
we define
$\|d_{\mathcal {L}}\|_{\mu ,B}$
to be the supremum of
$\text {dist}(\mathbf {y},\mathcal {L})$
over
$\mathbf {y} \in S \cap B,$
where
$\text {dist}(\mathbf {y},\mathcal {L})= \inf \{\|\mathbf {y}-\mathbf {l}\|: \mathbf {l} \in \mathcal {L}\} .$
Given an open set V of
$F^n$
and
$C,\alpha>0$
, we say that the measure
$\mu $
is
$(C,\alpha )$
-decaying on V if for any nonempty open ball
$B \subset V$
centred in
$S,$
any affine hyperplane
$\mathcal {L}$
of
$F^n$
and any
$\varepsilon>0$
,
$$ \begin{align*} \mu(B \cap \mathcal{L}^{(\varepsilon)}) \leq C \bigg(\frac{\varepsilon}{\|d_{\mathcal{L}}\|_{\mu,B}}\bigg)^{\alpha} \mu(B), \end{align*} $$
where
$\mathcal L^{(\varepsilon )}$
is the
$\varepsilon $
-neighbourhood of
$\mathcal L$
.
Definition 3.2. Let
$\mu $
be a nonatomic Borel measure on
$F^n.$
We say that the measure
$\mu $
is friendly if for
$\mu $
-almost every point
$\mathbf {y}_0 \in F^n$
, there exists a neighbourhood V of
$\mathbf {y}_0$
such that the restriction
$\mu |_V$
of
$\mu $
to V is finite, doubling, nonplanar and
$(C,\alpha )$
-decaying for some
$C,\alpha>0.$
Following [Reference Beresnevich and Velani4], the notion of d-contracting measure on
$F^n$
generalises the notion of contracting measure on
$F^n$
(identified with
$F^{1 \times n}$
or
$F^{n \times 1}$
).
Definition 3.3. Let
$d \in \mathbb N$
with
$0 \leq d \leq n-1.$
A finite, nonatomic and doubling Borel measure on
$F^n$
is said to be d-contracting if it satisfies the conditions (3.2) and (3.3) for a contracting measure with the plane
$\mathcal {L}_{\mathbf {b},\mathbf {c}}$
replaced by any d-dimensional plane
$\mathcal {L}.$
It is easy to see that a contracting measure on
$F^{1 \times n}$
is essentially the same as a
$0$
-contracting measure on
$F^n$
and a contracting measure on
$F^{n \times 1}$
is essentially the same as an
$(n-1)$
-contracting measure on
$F^n.$
Theorem 3.4. Let
$d \in \mathbb N $
with
$0 \leq d \leq n-1.$
Then any friendly measure on
$F^n$
is d-contracting almost everywhere.
Proof. Let
$d \in \mathbb N$
,
$0 \leq d \leq n-1 $
, and
$\mu $
be a friendly measure on
$F^{n}.$
Then for
$\mu $
-almost every point
$\mathbf {y}_0 \in F^n$
, there exists a neighbourhood V of
$\mathbf {y}_0$
such that the restriction
$\mu |_V$
of
$\mu $
to V is nonplanar, finite, doubling and
$(C,\alpha )$
-decaying on V for some
$C,\alpha>0.$
For the sake of simplicity, without loss of generality, we may assume that
$\mu =\mu |_V.$
We need to show that there exists
$C',\alpha ',r_0>0$
such that for any d-dimensional plane
$\mathcal {L}$
of
$F^n, 0 <\varepsilon < r_0 $
and
$0<\delta <1$
, the following property is satisfied: for all
$\mathbf {y} \in \mathcal {L}^{(\delta {\varepsilon })} \cap S$
, there is an open ball B centred at
$\mathbf {y}$
such that
$$ \begin{align} B \cap S \subset \mathcal{L}^{({\varepsilon})} \end{align} $$
and
$$ \begin{align} \mu(5B \cap \mathcal{L}^{(\delta {\varepsilon})}) \leq C' \delta^{\alpha'} \mu(5B ), \end{align} $$
where S is the support of
$\mu .$
First we observe that any d-dimensional plane of
$F^n$
is of the form
$$ \begin{align*} \mathcal L =\{\mathbf y=(y_1,\dots,y_n) \in F^n : b_{i1}y_1 + \dots+b_{in} y_n + a_i=0 \,\, \text{for} \,\, i=1,\dots,s\}, \end{align*} $$
where
$\mathbf b_i=(b_{i1},\dots ,b_{in}) \in F^n$
with
$\|\mathbf b_i\|=1$
and
$s \in \{1,\dots ,n\}.$
Then
$$ \begin{align*} \mathcal L^{(\varepsilon)} =\{\mathbf y=(y_1,\dots,y_n) \in F^n : |b_{i1}y_1 + \dots+b_{in} y_n + a_i|< \varepsilon \,\, \text{for} \,\, i=1,\dots,s\}. \end{align*} $$
Since
$\mu $
is nonplanar, the support S of
$\mu $
contains n linearly independent points
$\mathbf {y}_1,\dots , \mathbf {y}_n$
of
$F^n.$
Hence, we can find
$r_0>0$
such that whenever
$0< \varepsilon < r_0$
, the
$\varepsilon $
-neighbourhood of any d-dimensional plane
$\mathcal L$
cannot contain all the points
$\mathbf {y}_1,\dots , \mathbf {y}_n.$
Therefore, for any d-dimensional plane
$\mathcal L$
and
$0 < \varepsilon < r_0,$
$$ \begin{align} S \not\subset \mathcal{L}^{(\varepsilon)}. \end{align} $$
Fix
$0<\delta <1$
. If
$S \cap \mathcal {L}^{(\delta {\varepsilon })} = \emptyset ,$
then there is nothing to show. So we now assume that 
and let
$\mathbf {y} \in S \cap \mathcal {L}^{(\delta {\varepsilon })} .$
Our goal is to find a ball B centred at
$\mathbf {y}$
such that (3.4) and (3.5) hold. Since
$\mathcal {L}^{(\delta {\varepsilon })}$
is an open set, we can find a ball
$B'$
centred at
$\mathbf {y}$
such that
$$ \begin{align} B' \subset \mathcal{L}^{(\varepsilon)}. \end{align} $$
In view of (3.6) and (3.7), there exists a real number
$\gamma \geq 1$
such that
$$ \begin{align*} 5 \gamma B' \cap S \not\subset \mathcal{L}^{(\varepsilon)} \quad \text{and} \quad \gamma B' \cap S \subset \mathcal{L}^{(\varepsilon)}. \end{align*} $$
Hence, there exists a point
$\mathbf {y}'=(y_1',\dots ,y_n') \in (5 \gamma B' \cap S) \setminus \mathcal {L}^{(\varepsilon )} $
and this implies that
$$ \begin{align} |b_{l1}y_1' + \dots+b_{ln} y_n' + a_l| \geq \varepsilon \quad \text{for some} \,\, l \in \{1,\dots,s\}. \end{align} $$
Consider the hyperplane
$ \mathcal L_0 =\{\mathbf y=(y_1,\dots ,y_n) \in F^n : b_{l1}y_1 + \dots +b_{ln} y_n + a_l=0 \}$
. In view of (3.8) and using the ultrametric triangle inequality,
$\text {dist}(\mathbf {y}',\mathcal {L}_0) \geq \varepsilon .$
Take
$B=5\gamma B'.$
From the above discussion, it is clear that
$\|d_{\mathcal {L}_0}\|_{\mu ,B} \geq \varepsilon .$
Since
$\mu $
is
$(C,\alpha )$
-decaying with
$\delta \varepsilon $
in place of
$\varepsilon ,$
$$ \begin{align*} \mu(5\gamma B' \cap \mathcal{L}^{(\delta {\varepsilon})}) \leq \mu(5\gamma B' \cap \mathcal{L}_{0}^{(\delta {\varepsilon})}) \leq C \bigg(\frac{\delta \varepsilon}{\varepsilon} \bigg)^{\alpha} \mu(5 \gamma B')= C \delta^{\alpha} \mu(5\gamma B' ). \end{align*} $$
Therefore, the ball
$\gamma B'$
satisfies conditions (3.4) and (3.5) and
$\mu $
is d-contracting.
Corollary 3.5.
(1) Any friendly measure on
$F^{m \times 1}$
is (strongly) contracting almost everywhere.
(2) Any friendly measure on
$F^{1 \times n}$
is (strongly) contracting almost everywhere.
Proof. From Theorem 3.4, it follows at once that any friendly measure on
$F^{m \times 1}$
or
$F^{1 \times n}$
is contracting almost everywhere.
The proof that any friendly measure on
$F^{m \times 1}$
or
$F^{1 \times n}$
is strongly contracting almost everywhere is the same as the proof of Theorem 3.4.
4 Lower bounds for Diophantine exponents
For the nontrivial implication in Theorem 2.1(B), we need to show that a certain measure
$\mu $
on
$F^{m \times n}$
is inhomogeneously strongly extremal. This amounts to showing that the following two statements hold for all 
:
Proving the nontrivial implication in Theorem 2.1(A) amounts to showing analogous upper and lower bounds for 
We devote this section to prove (4.2), that is, the lower bound for inhomogeneous strong extremality. As a by-product, we get the lower bound for inhomogeneous extremality. Before going to the proof of this result, we collect some results related to the Diophantine transference principle over function fields. The first is the function field analogue of a result by Bugeaud and Laurent [Reference Bugeaud and Laurent5].
Theorem 4.1 (Bugeaud and Zhang, [Reference Bugeaud and Zhang6]).
Let
$Y \in F^{m \times n}$
. Then for all 
Furthermore, equality occurs in (4.3) for almost all 
.
Next, we recall a positive characteristic version of Dyson’s transference principle [Reference Dyson8].
Theorem 4.2 [Reference Ganguly, Ghosh, Moree, Pohl, Snoha and Ward10].
For any
$Y \in F^{m \times n},$
we have
$ \omega (Y) =1$
if and only if
$\omega (Y^t)=1$
.
Now we are ready to prove the desired lower bound.
Proposition 4.3. Let
$\mu $
be an extremal measure on
$F^{m \times n}.$
Then for every 
we have 
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
Proof. It is clear from the definitions that
for all
$Y \in F^{m \times n}$
and 
From Dirichlet’s theorem and the definition of uniform exponent,
$$ \begin{align} \hat{\omega}(Y) \geq 1 \quad\text{and}\quad \omega(Y) \geq \hat{\omega} (Y) \end{align} $$
for all
$Y \in F^{m \times n}.$
Since the given measure
$\mu $
is extremal,
$\omega (Y)=1$
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
From Theorem 4.2,
$\omega (Y^t)=1$
for
$\mu $
-almost every
$Y \in F^{m \times n}$
. Again from (4.5),
$\hat {\omega }(Y^t)=1$
for
$\mu $
-almost every
$Y \in F^{m \times n}$
. Finally, using Theorem 4.1,
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
This completes the proof in view of (4.4).
Proof of Proposition 1.8.
Let
$\mu $
be an inhomogeneously strongly extremal measure on
$F^{m \times n}.$
Given any 
, we have 
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
Therefore, 
for all 
and for
$\mu $
-almost every
$Y \in F^{m \times n}$
by (1.1).
To complete the proof, we need to show that 
for all 
and for
$\mu $
-almost every
$Y \in F^{m \times n}.$
Since
$\mu $
is inhomogeneously strongly extremal, it is trivially extremal. So the desired result follows from Proposition 4.3.
5 Inhomogeneous transference principle
The main ingredient in the proof of Theorems 2.1 and 2.4 is the inhomogeneous transference principle [Reference Beresnevich and Velani4, Section 5]. Let
$(X,d)$
be a locally compact metric space. Given two countable indexing sets
$\mathcal {A}$
and
$\mathbf {T},$
let H and I be two maps from
$\mathbf {T} \times \mathcal {A} \times \mathbb {R}_+$
into the set of open subsets of X such that
$$ \begin{align*} H : (\mathbf{t},\alpha, \lambda) \in \mathbf{T} \times \mathcal{A} \times \mathbb{R}_+ \mapsto H_{\mathbf{t}}(\alpha,\lambda) \end{align*} $$
and
$$ \begin{align*} I : (\mathbf{t},\alpha, \lambda) \in \mathbf{T} \times \mathcal{A} \times \mathbb{R}_+ \mapsto I_{\mathbf{t}}(\alpha,\lambda). \end{align*} $$
Also, let
$$ \begin{align*} H_{\mathbf{t}}(\lambda)\overset{\text{ def}}{=} \bigcup_{\alpha \in \mathcal{A}} H_{\mathbf{t}}(\alpha,\lambda) \quad \text{and} \quad I_{\mathbf{t}}(\lambda)\overset{\text{ def}}{=} \bigcup_{\alpha \in \mathcal{A}} I_{\mathbf{t}}(\alpha,\lambda). \end{align*} $$
Let
$\Phi $
denote a set of functions
$\phi : \mathbf {T} \rightarrow \mathbb {R}_+.$
For
$\phi \in \Phi ,$
consider the limsup sets
$$ \begin{align*} \Lambda_H(\phi) = \displaystyle \limsup_{\mathbf{t} \in \mathbf{T}} H_{\mathbf{t}}(\phi(\mathbf{t})) \quad \text{and} \quad \Lambda_I(\phi) = \displaystyle \limsup_{\mathbf{t} \in \mathbf{T}} I_{\mathbf{t}}(\phi({\mathbf{t}})). \end{align*} $$
We call the sets associated with the maps H and I the homogeneous and inhomogeneous sets, respectively. Now we discuss two important properties, which are the key ingredients for the inhomogeneous transference principle.
The intersection property. The triple
$(H,I,\Phi )$
satisfies the intersection property if for any
$\phi \in \Phi ,$
there exists
$\phi ^* \in \Phi $
such that for all but finitely many
$\mathbf {t} \in T$
, and all distinct
$\alpha $
and
$\alpha '$
in
$\mathcal {A},$
we have
$ I_{\mathbf {t}}(\alpha ,\phi (\mathbf {t})) \cap I_{\mathbf {t}}(\alpha ',\phi (\mathbf {t})) \subseteq H_{\mathbf {t}}(\phi ^*(\mathbf {t}))$
.
The contraction property. Let
$\mu $
be a nonatomic finite doubling measure supported on a bounded subset
$\mathbf {S}$
of
$X.$
We say
$\mu $
is contracting with respect to
$(I, \Phi )$
if for any
$\phi \in \Phi ,$
there exists
${\phi ^+ \in \Phi }$
and a sequence of positive numbers
$\{ k_{\mathbf {t}} \}_{\mathbf {t} \in \mathbf {T}}$
satisfying
$ {\sum _{\mathbf {t} \in \mathbf {T}} k_{\mathbf {t}} < \infty }$
, and such that for all but finitely many
$\mathbf {t} \in \mathbf {T}$
and all
$\alpha \in \mathcal {A}$
, there exists a collection
$C_{\mathbf {t},\alpha }$
of balls B centred in
$\mathbf {S}$
satisfying the following three conditions:
-
(C.1)
$\mathbf {S} \cap I_{\mathbf {t}}(\alpha ,\phi ({\mathbf {t}})) \subseteq \bigcup _{B \in C_{\mathbf {t},\alpha }} B$
; -
(C.2)
$\mathbf {S} \cap \bigcup _{B \in C_{\mathbf {t},\alpha }} B \subseteq I_{\mathbf {t}}(\alpha ,\phi ^+(\mathbf {t}))$
; and -
(C.3)
$\mu (5B \cap I_{\mathbf {t}}(\alpha ,\phi (\mathbf {t}))) \leq k_{\mathbf {t}} \mu (5B)$
.
Theorem 5.1 [Reference Beresnevich and Velani4, Theorem 5].
If the triple
$(H,I,\Phi )$
satisfies the intersection property and
$\mu $
is contracting with respect to
$(I,\Phi )$
, then
$$ \begin{align*} \mu(\Lambda_H(\phi))=0 \quad \text{for all } \phi \in \Phi \Longrightarrow \mu(\Lambda_I(\phi))=0 \quad \text{for all } \phi \in \Phi. \end{align*} $$
6 Upper bounds for Diophantine exponents
In this section, we will prove Theorem 2.1(B) (the proof of Theorem 2.1(A) is similar). Clearly, one direction is trivial. For the other direction, let
$\mu $
be a measure on
$F^{m \times n}$
, which is strongly contracting almost everywhere. We want to show that
$$ \begin{align*} \mu\ \text{is strongly extremal}\! \implies\! \mu\ \text{is inhomogeneously strongly extremal}. \end{align*} $$
Earlier we observed that this amounts to showing the upper and lower bounds, (4.1) and (4.2), respectively. We have already proved the lower bound in the last section. We prove the upper bound in this section using the inhomogeneous transference principle. In fact, we will prove the upper bound for the general nonextremal case (that is, Theorem 2.4(II)). We follow the strategy of Beresnevich and Velani [Reference Beresnevich and Velani4].
Let
$\mu $
be a measure on
$F^{m \times n},$
which is strongly contracting almost everywhere. Suppose that
$\omega ^{\times }(Y)=\eta $
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
We want to prove that, for all 
, 
for
$\mu $
-almost every
$Y \in F^{m \times n}.$
Define
Observe that Theorem 2.4(II) reduces to proving
Consider
$ \mathbf {T}=\mathbb Z^{m+n}$
. For each
$\mathbf {t}=(t_1,\dots ,t_{m+n})\in \mathbf {T},$
let
$$ \begin{align*} a_{\mathbf{t}}:={\operatorname{diag}} \{X^{t_1},\dots,X^{t_m},X^{-t_{m+1}},\dots,X^{-t_{m+n}}\} \in F^{m \times n}. \end{align*} $$
For any matrix
$Y \in F^{m \times n},$
let
$$ \begin{align*}U_Y :=\begin{bmatrix} I_m& Y\\0& I_n \end{bmatrix},\end{align*} $$
where
$I_k$
denotes the identity matrix of size
$k \times k.$
We can view
$U_{Y}$
as a linear operator on
$F^{m + n}.$
Now given any 
we define the affine transformation 
on
$F^{m + n}$
by
where
$\Theta =({\theta }_1,\dots ,{\theta }_m,0,\dots ,0)^t \in F^{m+n}.$
Let
$\mathcal {A}= \Lambda ^m \times (\Lambda ^n \setminus \{0\})$
. For
$\varepsilon>0, {\mathbf {t} \in T}$
and
${\alpha \in \mathcal {A},}$
let
and
For
$\tau>0,$
consider the function
$$ \begin{align} \phi^{\tau} : \mathbf{T} \rightarrow \mathbb R_{+} ;\,\, \mathbf{t} \mapsto \phi^{\tau}_{\mathbf{t}}:= e^{-\tau \sigma(\mathbf{t})}, \end{align} $$
where
$\sigma (\mathbf {t}):=t_1+\dots +t_{m+n},$
and the set
For 
that is, the homogeneous case, we denote 
by
$\Delta _{\mathbf {T}}(\phi ^{\tau }).$
The following proposition allows us to reformulate the set 
in terms of these limsup sets.
Proposition 6.1. There exists a subset
$\mathbf {T}$
of
$ \mathbb Z^{m + n}$
such that
$$ \begin{align} \sum_{\mathbf t \in \mathbf T} e^{-\tau \sigma(\mathbf t)} < \infty \quad \text{for all } \tau>0 \end{align} $$
and
Proof. First, we define the required set
$\mathbf T \subset \mathbb Z^{m+n}$
. For
$\mathbf {u}=(u_1,\dots ,u_m) \in \mathbb Z^{m}_{+}$
and
${\mathbf {v}=(v_1,\dots ,v_n) \in \mathbb Z^{n}_{+}}$
, we define
$$ \begin{align*} \sigma(\mathbf u):= \sum_{j=1}^{m} u_j, \quad \sigma(\mathbf v):= \sum_{i=1}^{n} v_i \quad\text{and}\quad \xi:=\xi(\mathbf u,\mathbf v)=\frac{\sigma(\mathbf u )- \eta \sigma(\mathbf v)}{m+\eta n}, \end{align*} $$
where
$\mathbb Z_{+}=\{s \in \mathbb Z: s \geq 0\}.$
Given
$\mathbf u$
and
$\mathbf v$
as above, define the
$(m+n)$
-tuple
${\mathbf {t}=(t_1,\dots ,t_{m+n})}$
as
$$ \begin{align} \mathbf{t}:=(u_1-[\xi],\dots,u_m - [\xi], v_1 + [\xi] , \dots, v_n +[\xi]), \end{align} $$
where for any
$x \in \mathbb R, [x]$
denotes the greatest integer not greater than
$x.$
Finally,
$$ \begin{align*} \mathbf T := \{\mathbf t \in \mathbb Z^{m+n} \,\, \text{given by} \,\, (6.6): \mathbf u \in \mathbb Z^{m}_{+}, \,\, \mathbf{v} \in \mathbb Z^{n}_{+} \,\, \text{with} \,\, \sigma(\mathbf u) \geq \eta \sigma(\mathbf v)\}. \end{align*} $$
We show that
$\mathbf T$
has the properties required in Proposition 6.1. First, let us record a few inequalities which will be essential later. Note that
$$ \begin{align} \sigma(\mathbf t) := \sum_{i=1}^{m+n} t_i = \sigma(\mathbf u) - m [\xi]+\sigma(\mathbf v)+ n[\xi] = \frac{\eta +1}{\eta} \sigma(\mathbf u)-m\bigg(\frac{\xi}{\eta} + [\xi] \bigg)+n([\xi]- \xi) \end{align} $$
and also
$$ \begin{align} \sigma(\mathbf t)= (\eta+1) \sigma(\mathbf v) +m(\xi -[\xi])+n([\xi]+\eta \xi ). \end{align} $$
From (6.7) and (6.8) respectively
$$ \begin{align} \frac{\eta}{\eta+1} \sigma(\mathbf t) & = \sigma(\mathbf u)- \frac{m \eta}{\eta +1}\bigg(\frac{\xi}{\eta} + [\xi] \bigg)+\frac{n \eta}{\eta +1}([\xi]- \xi) \nonumber \\ & \leq \sigma(\mathbf u)- \frac{m \eta}{\eta +1}\bigg(\frac{\xi}{\eta}+ [\xi] \bigg), \quad \text{since} \,\, \xi-1 < [\xi] \leq \xi \nonumber \\ & = \sigma(\mathbf u)- \frac{m \eta}{\eta +1}\bigg(\frac{[\xi]}{\eta}+ [\xi] \bigg) + \frac{m \eta}{\eta +1} \bigg(\frac{[\xi]}{\eta}-\frac{\xi}{\eta} \bigg) \nonumber \\ & \leq \sigma(\mathbf{u}) - m [\xi], \quad \text{since} \,\, \xi-1 < [\xi] \leq \xi \end{align} $$
and
$$ \begin{align} \frac{1}{\eta+1}\sigma(\mathbf t) &= \sigma(\mathbf v) +\frac{m}{\eta+1}(\xi -[\xi])+\frac{n}{\eta+1}([\xi]+\eta \xi ) \nonumber \\ & \geq \sigma(\mathbf v) +\frac{n}{\eta+1}([\xi]+\eta \xi ), \quad \text{since } \xi-1 < [\xi] \leq \xi \nonumber \\ &= \sigma(\mathbf v) +\frac{n}{\eta+1}([\xi]+\eta [\xi] )+\frac{n}{\eta+1} \left(\eta \xi-\eta[\xi]\right) \nonumber \\ & \geq \sigma(\mathbf{v}) + n [\xi], \quad \text{as} \,\, \xi-1 < [\xi] \leq \xi. \end{align} $$
Since
$\xi \geq 0,$
in view of (6.9) and (6.10),
$$ \begin{align} (\eta+1)\sigma(\mathbf v) \leq \sigma(\mathbf t) \leq \frac{\eta+1}{\eta}\sigma(\mathbf u). \end{align} $$
From (6.9), (6.10) and using
$\xi -1 < [\xi ] \leq \xi $
,
$$ \begin{align} \sigma(\mathbf t)= \frac{\eta}{\eta+1} \sigma(\mathbf{t})+ \frac{1}{\eta+1} \sigma(\mathbf{t}) & =\sigma(\mathbf{u}) + \sigma(\mathbf{v}) - (m-n) [\xi] \nonumber \\ & \geq \sigma(\mathbf{u}) + \sigma(\mathbf{v}) - (m-n) \xi \nonumber \\ & = \sigma(\mathbf{u}) + \sigma(\mathbf{v}) - \frac{(m-n)}{m+\eta n} (\sigma(\mathbf{u}) - \eta \sigma(\mathbf{v}) ) \nonumber \\ & = \frac{\eta +1}{m+\eta n}(n \sigma(\mathbf{u}) + m \sigma(\mathbf{v})). \end{align} $$
From (6.12), we will conclude that the series (6.5) is convergent and, for any
$r \in \mathbb R_+$
,
$$ \begin{align} \# \{ \mathbf t \in \mathbf T : \sigma(\mathbf t) < r\} < \infty. \end{align} $$
First, we show that
Let 
Note that 
if and only if there exists
$\varepsilon>0$
such that for arbitrarily large
$T \geq 1$
, there exists
$\alpha =(\mathbf {p}, \mathbf q) \in \mathcal A=\Lambda ^m \times (\Lambda ^n \setminus \{0\})$
satisfying 
such that
Since 
(6.15) is satisfied for infinitely many
$T\in \mathbb N$
. For any such
$T ,$
there is a unique
$\mathbf {u}=(u_1,\dots ,u_m) \in \mathbb Z^{m}_{+}$
and
$\mathbf {v}=(v_1,\dots ,v_n) \in \mathbb Z^{n}_{+}$
such that
$$ \begin{align} e^{-u_j} \leq \max \{ |Y_j \mathbf q +p_j +{\theta}_j|, e^{-(\eta+\varepsilon)T} \} < e^{-u_j+1} \quad \text{for } 1 \leq j \leq m \end{align} $$
and
$$ \begin{align} e^{v_i} \leq \max \{1,|q_i|\} < e^{v_i +1} \quad \text{for } 1 \leq i \leq n, \end{align} $$
where
$Y_j$
denotes the jth row of
$Y \in F^{m \times n}.$
From (6.16) and (6.17),
Now (6.15) and (6.18) imply that
$e^{-\sigma (\mathbf u)} < e^{-\sigma (\mathbf v)(\eta +\varepsilon )}.$
Therefore,
$$ \begin{align} \sigma(\mathbf u) - \eta \sigma(\mathbf v)> \varepsilon \sigma(\mathbf v) \geq 0. \end{align} $$
Hence,
$\mathbf t$
given by (6.6) with
$\mathbf u=(u_1,\dots ,u_m)$
and
$\mathbf v=(v_1,\dots ,v_n)$
satisfying (6.16) and (6.17), respectively, is in
$\mathbf T. $
If
$\sigma (\mathbf u) \leq 2 \eta \sigma (\mathbf v),$
then from (6.11) and (6.19),
$$ \begin{align*} \xi=\frac{\sigma(\mathbf u )- \eta \sigma(\mathbf v)}{m+\eta n} \geq \frac{\varepsilon \sigma(\mathbf v)}{m+\eta n} \geq \frac{\varepsilon \sigma(\mathbf u)}{2 \eta (m+n)}> \frac{\varepsilon \sigma (\mathbf t)}{2(\eta +1)(m+n)}. \end{align*} $$
If
$\sigma (\mathbf u)> 2 \eta \sigma (\mathbf v),$
then from (6.11),
$$ \begin{align*} \xi=\frac{\sigma(\mathbf u )- \eta \sigma(\mathbf v)}{m+ \eta n} = \frac{2\sigma(\mathbf u )- 2\eta \sigma(\mathbf v)}{2(m+ \eta n)}> \frac{ \sigma(\mathbf u)}{2(m+\eta n)} > \frac{\eta \sigma (\mathbf t)}{2(\eta +1)(m+ \eta n)}. \end{align*} $$
In view of these two inequalities,
$$ \begin{align} \xi> \tau_0 \sigma(\mathbf t),\quad \text{where } \tau_0:=\frac{ \min\{\varepsilon,\eta \}}{2(\eta +1)(m+\eta n)}. \end{align} $$
Now we take
$$ \begin{align*} a_{\mathbf t}=X^{-([\xi]+1)} {\operatorname{diag}} \{X^{u_1},\dots,X^{u_m},X^{-v_1},\dots,X^{-v_n}\}. \end{align*} $$
Combining (6.20) and (6.21), we conclude that for
$0<\tau <\tau _0$
,
for all sufficiently large
$\sigma (\mathbf t).$
Together, (6.15) and (6.16) imply that
$\sigma (\mathbf u) \rightarrow \infty $
as
${T \rightarrow \infty .}$
Hence, in view of (6.12) and the fact that (6.15) holds for arbitrarily large
$T \in \mathbb N,$
we conclude that (6.22) holds for infinitely many
$\mathbf t \in \mathbf T.$
Therefore, 
for any
$\tau \in (0,\tau _0).$
This completes the proof of (6.14).
Finally, to complete the proof of Proposition 6.1, we show that
Let 
for some
$\tau>0. $
By definition, 
for infinitely many
$\mathbf t \in \mathbf T.$
For any such
$\mathbf t,$
there exists
$\alpha =(\mathbf p, \mathbf q) \in \mathcal A$
such that 
. By taking the product of the first m coordinates and the last n nonzero coordinates of 
, respectively we get
$$ \begin{align*} \prod_{j=1}^{m} e^{t_j}|Y_j \mathbf q + p_j+{\theta}_j| < e^{-m \tau \sigma(\mathbf t)} \quad\text{and}\quad \prod_{1 \leq i \leq n, \, q_i \neq 0} e^{-t_{m+i}} |q_i| < e^{- n \tau \sigma(\mathbf t)}. \end{align*} $$
From (6.11) and the fact that
$\sigma (\mathbf {t}) \geq 0$
(by (6.10)),
and
$$ \begin{align} {\prod}_+ (\mathbf q) < e^{- n \tau \sigma(\mathbf t)} e^{ \sigma(\mathbf t)/{(\eta+1)}} < e^{ \sigma(\mathbf t)/{(\eta+1)}}. \end{align} $$
If we take
$T= \sigma (\mathbf t)/{(\eta +1)}$
and
$\varepsilon :=m \tau (\eta +1),$
then (6.23) and (6.24) hold for arbitrarily large
$T.$
Therefore, 
, completing the proof of Proposition 6.1.
Proof of Theorem 2.4.
In view of (6.1) and Proposition 6.1, we first note that to prove Theorem 2.4(II), it is enough to show that
We use the inhomogeneous transference principle to prove (6.25). From now on, 
is fixed and, without loss of generality, we assume that
$\mu $
is strongly contracting on
$F^{m \times n}.$
Let
$X:=F^{m \times n}$
,
$\mathcal A= \Lambda ^m \times (\Lambda ^n \setminus \{0\}), \mathbf T$
be as in Proposition 6.1, and the maps H and I be given by
where
$\varepsilon> 0 , \mathbf {t} \in \mathbf T$
,
$\alpha \in \mathcal A$
and 
is given by (6.2). From these definitions, it follows readily that 
and 
Let
$\Phi $
be the collection of functions given by (6.3). Then
where 
is given by (6.4). Now (6.25) will immediately follow by applying the inhomogeneous transference principle if we can verify that the triple
$(H,I,\Phi )$
satisfies the intersection property and the measure
$\mu $
is contracting with respect to
$(I,\Phi ).$
The intersection property. Let
$\alpha =(\mathbf {p},\mathbf {q})$
and
$\alpha '=(\mathbf {p}',\mathbf {q}')$
be two distinct elements in
$\mathcal {A},$
where
$\mathbf {p},\mathbf {p}' \in \Lambda ^m$
and
$\mathbf {q},\mathbf {q}' \in \Lambda ^n \setminus \{0\}$
. Also let
$\phi \in \Phi .$
Then
$\phi (\mathbf t)=e^{-\tau \sigma (\mathbf t)}$
for some
$\tau>0.$
Consider any element Y from
$I_{\mathbf {t}}(\alpha ,\phi (\mathbf {t})) \cap I_{\mathbf {t}}(\alpha ',\phi (\mathbf {t})) .$
Then
Since
$\mathbf T$
satisfies (6.5) and (6.13),
for all but finitely many
$\mathbf {t} \in \mathbf T.$
Let
$\alpha ":=\alpha -\alpha '=(\mathbf {p}",\mathbf {q}"),$
where
$\mathbf {p}"=\mathbf {p}-\mathbf {p'} \in \Lambda ^m$
and
$\mathbf {q}"=\mathbf {q}-\mathbf {q}' \in \Lambda ^n.$
If
$\mathbf q"=0,$
from (6.26), we get
$\|\mathbf p"\| < 1$
for all but finitely many
$\mathbf t \in \mathbf T .$
Then
$\mathbf p"=0$
(since
$\mathbf {p"} \in \Lambda ^m$
), which is a contradiction as
$\alpha \neq \alpha '.$
Hence,
$\mathbf q" \neq 0$
and so
$\alpha " \in \mathcal A$
. Therefore,
$Y \in \Delta _{\mathbf t}(\alpha ", \phi (\mathbf t)) \subset H_{\mathbf {t}}(\phi (\mathbf t)).$
This completes the verification of the intersection property with
$\phi ^*=\phi \in \Phi .$
The contracting property. Recall that
$\mu $
is a strongly contracting measure on
$F^{m \times n}.$
Without loss in generality, we may assume that the support S of
$\mu $
is bounded. Note that
$\mu $
is already doubling, finite and nonatomic. Hence, to show that
$\mu $
is contracting with respect to
$(I,\Phi ),$
it only remains to verify conditions (C.1)–(C.3). Let
$\phi \in \Phi .$
Then
$\phi (\mathbf t)=e^{- \tau \sigma (\mathbf t)}$
for some constant
$\tau>0.$
Define
$\phi ^+ :=\sqrt {\phi } \in \Phi .$
Let
$r_0$
be as in the definition of strongly contracting measure. Since
$\mathbf T$
satisfies (6.5) and (6.13),
$$ \begin{align} \phi^+(\mathbf t) \leq \min \{1,r_0\} \quad \text{and} \quad \sigma(\mathbf t) \geq 0 \end{align} $$
for all but finitely many
$\mathbf t \in \mathbf T.$
If
$\mathbf t=(t_1,\dots ,t_{m+n}) \in \mathbf T$
and
$\alpha '=(\mathbf p',\mathbf q') \in \mathcal A,$
the set
$I_{\mathbf t}(\alpha ', \phi (\mathbf t))$
is essentially the set of all
$Y \in F^{m \times n}$
such that
$$ \begin{align} |Y_j \mathbf q'+p_j'+{\theta}_j| < e^{-t_j} \phi(\mathbf t) \quad \text{and} \quad |q_i'| < e^{t_{m+i}} \phi(\mathbf t) \end{align} $$
for
$1 \leq j \leq m$
and
$1 \leq i \leq n.$
In a similar fashion, we see that
$I_{\mathbf t}(\alpha ',\phi ^+(\mathbf t))$
is the set of all
$Y \in F^{m \times n}$
such that
$$ \begin{align} |Y_j \mathbf q'+p_j'+{\theta}_j| < e^{-t_j} \phi^+(\mathbf t) \quad \text{and} \quad |q_i'| < e^{t_{m+i}} \phi^+(\mathbf t). \end{align} $$
Now we define
$$ \begin{align*} \varepsilon_j=\varepsilon_{j,\mathbf t} := \frac{e^{-t_j} \phi^+(\mathbf t)}{\|\mathbf q'\|} \quad \text{for } j=1,\dots,m \quad \text{and} \quad \delta=\delta_{\mathbf t} := \phi^+(\mathbf t). \end{align*} $$
Equation (6.11) and
$\sigma (\mathbf t) \geq 0$
imply that
$\sum _{j=1}^{m} t_j \geq 0.$
Hence, there exists some
${ l \in \{1,\dots , m\}}$
such that
$t_l \geq 0.$
Therefore, since
$\|\mathbf q'\| \geq 1$
and using (6.27),
$$ \begin{align*} \min_{1 \leq j \leq m} \varepsilon_{j,\mathbf t} < r_0 \quad \text{and} \quad \delta_{\mathbf t} <1. \end{align*} $$
Note that
$\delta \varepsilon _j={e^{-t_j}\phi (\mathbf t)}/{\|\mathbf q'\|}$
for all
$j=1,\dots ,m.$
Let 
and 
be given by (3.1) with
$\mathbf c:= X^{-(\max _{1 \leq i \leq n}\deg q_i')} \mathbf q'$
and 
where
$\mathbf q'=(q_1',\dots ,q_n').$
Then in view of (6.28) and (6.29),
Since
$\mu $
is strongly contracting, for all 
, there is an open ball
$B_Y$
centred at Y satisfying (3.2) and (3.3). Following the notation from Section 5, define
$C_{\mathbf t,\alpha '}$
to be the collection of all such balls. Also, define
$$ \begin{align*} k_{\mathbf t}:= C (\phi^+)^{\alpha}, \end{align*} $$
where C and
$\alpha $
are constants as in the definition of strongly contracting measure. By (6.5),
$\sum _{\mathbf t \in \mathbf T} k_{\mathbf t} < \infty .$
Observe that condition (C.1) follows from the definition of
$C_{\mathbf t, \alpha '}.$
Finally, conditions (C.2) and (C.3) follow from (3.2), (3.3) and (6.30). This shows that
$\mu $
is contracting with respect to
$(I, \Phi ).$
Remark 6.2. To prove Theorem 2.4(I), we take 
and show that 
for all 
Proposition 6.1 remains unchanged and the only change needed in the proof of Proposition 6.1 is the definition of
$\mathbf T.$
For
$u \in \mathbb Z_+$
and
$v \in \mathbb Z_+,$
let
$\mathbf u:=(u,\dots ,u) \in \mathbb Z_+^m$
,
$\mathbf v=(v_1,\dots ,v_n) \in \mathbb Z_+^n$
and
${\xi ={(mu-\eta nv)}/{(m+\eta n)}}.$
Given u and v, define
$\mathbf t=(t_1,\dots ,t_{m+n})$
by
$$ \begin{align} \mathbf t :=(u-[\xi],\dots,u-[\xi],v+[\xi],\dots,v+[\xi]). \end{align} $$
Define
$$ \begin{align*} \mathbf{T}:=\{\mathbf t \in \mathbb Z^{m+n} \,\, \text{given by} \,\, (6.31): u,v \in \mathbb Z_+ \,\, \text{with} \,\, mu \geq \eta n v \}. \end{align*} $$
The new version of Proposition 6.1 follows by appropriately modifying the arguments of the proof of Proposition 6.1. Once we have Proposition 6.1, we can apply the inhomogeneous transference principle to get our desired result. This also completes the proof of Theorem 2.1(A).
Acknowledgements
We are indebted to the Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, for providing a research-friendly atmosphere. S.D. is also thankful to the International Centre for Theoretical Sciences (ICTS), as a significant part of this project was completed during a visit (ICTS/etds-2022/12) to ICTS. Finally, we thank the anonymous referee for careful reading of this paper and helpful suggestions, which have improved the presentation of this paper.
