On the Hausdorff dimension of Weighted Singular Vectors

Gaurav Aggarwal Gaurav Aggarwal
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India 400005 [email protected] and Anish Ghosh Anish Ghosh
School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India 400005 [email protected]

Abstract.

We prove a sharp upper bound on the Hausdorff dimension of weighted singular vectors in $\mathbb{R}^{m}$ using dynamics on homogeneous spaces, specifically the method of integral inequalities. Together with the lower bound proved recently by Kim and Park [KimPark2024], this determines the Hausdorff dimension of weighted singular vectors, thereby generalizing to arbitrary dimension, the work of Liao, Shi, Solan, and Tamam [LSST], who determined the Hausdorff dimension of weighted singular vectors in two dimensions. We also provide the first known bounds for the Hausdorff dimension of weighted singular vectors restricted to fractal subsets.

Key words and phrases:

Diophantine approximation, ergodic theory, Hausdorff dimension, flows on homogeneous spaces

2020 Mathematics Subject Classification:

11J13, 11J83, 37A17

A. Ghosh gratefully acknowledges support from a grant from the TNQ foundation under the “Numbers and Shapes” initiative and an endowment from the Infosys foundation. G. Aggarwal and A. Ghosh gratefully acknowledges a grant from the Department of Atomic Energy, Government of India, under project

12-R\&D-TFR-5.01-0500

1. Introduction

Singular vectors are a class of vectors for which Dirichlet’s theorem in Diophantine approximation can be ‘infinitely improved’. More precisely,

Definition 1.1.

Fix $m\in\mathbb{N}$ and $w=(w_{1},\ldots,w_{m})\in\mathbb{R}^{m}$ such that $w_{1}\geq w_{2}\geq\ldots\geq w_{m}>0$ and $w_{1}+\ldots+w_{m}=1$ . A vector $\theta=(\theta_{1},\ldots,\theta_{m})\in\mathbb{R}^{m}$ is called $w$ -singular if for any $\varepsilon>0$ , there exists $T_{\varepsilon}>0$ such that for all $T>T_{\varepsilon}$ the following system of inequalities

	$\displaystyle\|p_{i}+q\theta_{i}\|$	$\displaystyle\leq\frac{\varepsilon}{T^{w_{i}}}\quad\text{ for all }i=1,\ldots,m$
	$\displaystyle 0$	$\displaystyle<q<T,$

has an integer solution $(p,q)\in\mathbb{Z}^{m}\times\mathbb{N}$ . We denote by $\text{Sing}(w)\subset\mathbb{R}^{m}$ , the set of all $w$ -singular vectors in $\mathbb{R}^{m}$ .

While the study of the classical ‘non-weighted’ situation, i.e. the study of $\text{Sing}(1/m,\dots,1/m)$ is more ubiquitous, Diophantine approximation with weights has been extensively studied in recent times. We refer the reader to [CGGMS] for an introduction. The set of singular vectors is clearly non-empty and due to a classical result of Khintchine, it contains uncountably many vectors when the dimension is greater than $1$ . Moreover, a modification of a classical argument in [Casselsbook] shows that the set of singular vectors has zero Lebesgue measure. There is a closely related notion of Dirichlet improvable vectors, see [KleinbockRao] for the relation between singular vectors with Dirichlet improvable vectors with respect to different norms.

Two central questions about singular vectors have recently been the focus of several important works:

(1)

to estimate the measure of $\text{Sing}(w)$ with respect to other natural measures. This problem encapsulates the themes of Diophantine approximation on manifolds as well as that on fractals,
(2)

to estimate the Hausdorff dimension of $\text{Sing}(w)$ , again one could ask for the Hausdorff dimension of $\text{Sing}(w)$ restricted to a manifold or a fractal.

Following work of several authors, an approach to question 1 was developed by Kleinbock and Weiss in [KWsingular] where it was shown that if $\mu$ belongs to a class of measures called friendly measures, then $\mu(\text{Sing}(w))=0$ . Friendly measures form a reasonably large class of measures and include some IFS’s as well as pushforwards of Lebesgue measure by smooth maps. There has been recent progress in showing the existence of weighted singular vectors on manifolds, cf. [KMWW, DattaTamam].

The second question, i.e. of estimating the Hausdorff dimension of $\text{Sing}(w)$ turns out to be a difficult problem and has received much attention in the last two decades. In a landmark work [Cheung], Y. Cheung showed that the dimensionof $\text{Sing}(1/2,1/2)\subset\mathbb{R}^{2}$ is $4/3$ . This was subsequently generalized to $\mathbb{R}^{n}$ in an important work of Cheung and Chevallier [CheungChevallier]. Another important result was obtained by Kadyrov, Kleinbock, Lindenstrauss and Margulis in [KKLM] using methods from homogeneous dynamics. Namely, a sharp upper bound on the more general set of singular on average $m\times n$ matrices was obtained using integral inequalities as introduced in the famous work [EMM] on the Oppenheim conjecture. The complementary lower bound was obtained by Das, Fishman, Simmons and Urbanski in [DFSU] using methods from the parametric geometry of numbers, see also the recent paper [Solan] for more results in this direction. In [Khalilsing], Khalil upgraded the integral inequality approach of [KKLM] using a beautiful argument to deal with singular vectors on fractals. That is, an upper bound on the Hausdorff dimension of singular vectors lying on self-similar fractals in $\mathbb{R}^{m}$ that satisfied the open set condition was obtained. In [ShahYang], Shah and Yang obtained bounds for the dimension of certain singular vectors lying on affine subspaces.

All the impressive results towards question 2 discussed above are in the ‘unweighted’ setting. The literature in the more complicated weighted setting is sparser. In [LSST], the Hausdorff dimension of $\text{Sing}(w)\subset\mathbb{R}^{2}$ was shown to be equal to $2-\frac{1}{1+w_{1}}$ . As far as higher dimensions are concerned, the authors (page 836 in [LSST]) write that “it is likely” that the dimension of weighted singular vectors in $\mathbb{R}^{m}$ is $m-1/\lambda_{1}$ “where $\lambda_{1}$ is the top Lyapunov exponent for the adjoint action of the corresponding one-parameter semigroup on the corresponding unipotent group.”

This expectation turns out to be accurate. The lower bound was recently obtained by Kim and Park in [KimPark2024] using an appropriate generalization of the technique of [LSST] that involves the construction of a fractal set contained in $\text{Sing}(w)$ . A corollary of our main result provides the matching upper bound, and so together we have

Theorem 1.2.

Fix $m\in\mathbb{N}$ and $w=(w_{1},\ldots,w_{m})\in\mathbb{R}^{m}$ such that $w_{1}\geq w_{2}\geq\ldots\geq w_{m}>0$ and $w_{1}+\ldots+w_{m}=1$ . Then the Hausdorff dimension of $\text{Sing}(w)$ satisfies

\displaystyle\dim_{H}(\text{Sing}(w))=m-\frac{1}{1+w_{1}}.

(1)

In fact, our methods also allow us to deal with Diophantine approximation on fractals; providing the very first dimension bounds for weighted singular vectors on fractals. Here is another corollary of our main theorem.

Theorem 1.3.

Fix $m\in\mathbb{N}$ . Fix $w=(w_{1},\ldots,w_{m})\in\mathbb{R}^{m}$ such that $w_{1}\geq w_{2}\geq\ldots\geq w_{m}>0$ and $w_{1}+\ldots+w_{m}=1$ . For each $1\leq i\leq m$ , let $\Phi_{i}$ be an IFS consisting of contractive similarities with equal contraction ratios on $\mathbb{R}$ and satisfying the open set condition. Let $\mathcal{K}_{i}$ be the limit set of $\Phi_{i}$ and $\mathcal{K}=\prod_{i}\mathcal{K}_{i}$ . Then, the dimension of $\text{Sing}(w)\cap\mathcal{K}$ satisfies

\displaystyle\dim_{H}(\text{Sing}(w)\cap\mathcal{K})\leq\dim_{H}(\mathcal{K})-\frac{\min_{i}(\dim_{H}\mathcal{K}_{i})}{1+w_{1}},

(2)

where $s=dim(\mathcal{K})$ .

We will now state our main theorem, postponing some notation for later. Our approach is dynamical in nature and owes its existence to an influential work of Dani [Dani-Crelle] where a connection was established between Diophantine properties of vectors and dynamical and topological properties of certain orbits of an associated unimodular lattice. Let $\mathcal{X}=\text{SL}_{m+1}(\mathbb{R})/\text{SL}_{m+1}(\mathbb{Z})$ be the space of unimodular lattices in $\mathbb{R}^{n}$ . For $t>0$ and $\theta\in\mathbb{R}^{m}$ , define

\displaystyle g_{t}=\begin{pmatrix}t^{w_{1}}\\ &\ddots\\ &&t^{w_{m}}\\ &&&t^{-1}\end{pmatrix},\quad u(\theta)=\begin{pmatrix}I_{m}&\theta\\ &1\end{pmatrix}.

(3)

For $x\in\mathcal{X}$ define the set

\text{Div}(x,w):=\{\theta\in\mathbb{R}^{m}~{}:~{}g_{t}u(\theta)x\rightarrow\infty\}.

By Dani’s results mentioned above (the famous “Dani correspondence”), singular vectors correspond exactly to the above set for $x=e\Gamma$ . The main result of the paper is the following theorem.

Theorem 1.4.

\displaystyle\dim_{H}(\text{Div}(x,w)\cap\mathcal{K})\leq\dim_{H}(\mathcal{K})-\frac{\min_{i}(\dim_{H}\mathcal{K}_{i})}{1+w_{1}}.

(4)

1.1. Outline of proof

The core strategy of the proof closely mirrors the approach developed in [KKLM]. The key idea is to construct a family of height functions that satisfy the contraction hypothesis. These height functions quantify the depth of orbits as they approach the cusp. Once such a family is constructed, it is then used iteratively to generate an open covering of $\text{Div}(x,w)$ . Specifically, by applying the contraction hypothesis iteratively, we construct coverings of $\text{Div}(x,v)$ using balls of nearly uniform diameter, with the number of balls being bounded by a function of their common diameter. This estimate ensures the finiteness of the $l$ -dimensional Hausdorff measure of $\mathcal{K}$ for specific values of $l$ smaller than $d$ , thereby providing an upper bound on $\text{Div}(x,w)$ . This bound corresponds to the one stated in Theorem 1.4.

Despite its apparent simplicity, the approach presents significant challenges. The biggest one is constructing a family of height functions that satisfy the optimal contraction conditions. The optimality is important because the bound for $\text{Div}(x,w)$ depends very sensitively on the contraction rate of the height functions. Existing techniques can create height functions, but they aren’t optimal unless all the weights are equal. As a result, the best possible bound for $\mathbb{R}^{m}$ , without the new tools in this paper, is $m-\frac{mw_{m}}{1+w_{1}}$ , which only matches the actual bound when $mw_{m}=1$ , meaning all the weights are equal (where $w_{1}\geq w_{2}\geq\dots\geq w_{m}$ ).

To address this, we introduce some new concepts. First, we redefine the covolume of sublattices to depend on weights, as done in Section 3. Using this new definition of covolume $\|.\|$ , we as usual define the following proper functions on $\mathcal{X}$ :

\varphi_{l}:\mathcal{X}\rightarrow\mathbb{R},\quad\Lambda\mapsto\max\left\{\frac{1}{\|\Lambda_{l}\|}:\Lambda_{l}\text{ is a primitive sublattice of $\Lambda$ of rank $l$}\right\}.

In the unweighted case, the linear combination of these $\varphi_{l}$ acts as a height function, and one of the essential ingredients in proving that the linear combination is indeed a height function is a result from [EMM], Lemma 5.6, which states that for any two sublattices $\Lambda_{1},\Lambda_{2}$ of $\Lambda$ and with usual definition of covolume $\|.\|$ , we have

\|\Lambda_{1}\cap\Lambda_{2}\|\|\Lambda_{1}+\Lambda_{2}\|\leq\|\Lambda_{1}\|\|\Lambda_{2}\|.

This inequality is used to link the average rate of expansion of vectors in certain representations to recurrence results on $\mathcal{X}$ , which leads to the height function. The most crucial step of the paper is to prove an analogue of this result using the redefined covolume. Although this may seem straightforward, the new covolume definition introduces complications. For instance, it is not even true that for any $g\in G$ , one can find a constant $C_{g}>0$ such that

\|g\Lambda\|\leq C_{g}\|\Lambda\|,\quad\text{for all rank }l\text{ sublattices }\Lambda\text{ in }\mathbb{R}^{m+1}.

The analogous key inequality is referred to as the “weighted Mother inequality” and is discussed in Section 3. Section 4 proves the optimal expansion rate of vectors in the exterior powers of the standard representation of $\text{SL}_{m+1}(\mathbb{R})$ with respect to the quasinorm based on weights. Section 5 constructs the height functions using results from Section 4 and the weighted Mother inequality.

The second challenge lies in deriving dimension bounds from the contraction hypothesis. This is done iteratively. In the equal weight case, for $[0,1]^{m}$ , the process is as follows: Start with $[0,1]^{m}$ and divide it into $2^{m}$ parts, i.e., $[i_{1}/2,(i_{1}+1)/2]\times\dots\times[i_{m}/2,(i_{m}+1)/2]$ . Applying the contraction hypothesis with the base point $x$ , at most $2^{\gamma}$ intervals are needed to cover $\text{Div}(x,w)$ .

Now, pick one of the $\ell_{\infty}$ -balls that intersect $\text{Div}(x,w)$ , say $[i_{1}/2,(i_{1}+1)/2]\times\dots\times[i_{m}/2,(i_{m}+1)/2]$ , and divide it into $2^{m}$ smaller parts. By applying the contraction hypothesis again with base point $g_{2^{m/(1+m)}}u(i_{1}/2,\dots,i_{m}/2)x$ , and using the fact that

	$\displaystyle\{g_{2^{m/(1+m)}}u(\theta)g_{2^{m/(1+m)}}u(i_{1}/2,\dots,i_{m}/2):\theta\in[0,1]^{m}\}$
	$\displaystyle=\{g_{2^{2m/(1+m)}}u(\theta):\theta\in[i_{1}/2,(i_{1}+1)/2]\times\dots\times[i_{m}/2,(i_{m}+1)/2]\}$

and that we know the point $g_{2^{2m/(1+m)}}u(i_{1}/2,\dots,i_{m}/2)x$ is near the cusp (since $\text{Div}(x,w)\cap[i_{1}/2,(i_{1}+1)/2]\times\dots\times[i_{m}/2,(i_{m}+1)/2]\neq\emptyset$ ), we conclude that at most $2^{\gamma}$ intervals are needed to cover $\text{Div}(x,w)\cap[i_{1}/2,(i_{1}+1)/2]\times\dots\times[i_{m}/2,(i_{m}+1)/2]$ . Thus, $2^{2\gamma}$ $\ell_{\infty}$ -balls of diameter $1/2^{2}$ are needed to cover $\text{Div}(x,w)$ . Repeating this process iteratively, we find that $2^{k\gamma}$ $\ell_{\infty}$ -balls of diameter $1/2^{k}$ are needed to cover $\text{Div}(x,w)$ , which gives the bound $\dim_{H}(\text{Div}(x,w))\leq\gamma$ .

In the equal weight case, this process is simpler because the action of $g_{t}$ (as defined in (3) for $w_{1}=\dots=w_{m}$ ) expands the interval $[0,1]^{m}$ evenly. This allows us to easily subdivide it into cubes like $[i_{1}/2^{N},(i_{1}+1)/2^{N}]\times\dots\times[i_{m}/2^{N},(i_{m}+1)/2^{N}]$ and conjugate them with $g_{2^{m/(m+1)}}$ , making the integral over these cubes behave like that over $[0,1]^{m}$ . However, in the unequal expansion case, such simple divisions of $[0,1]^{m}$ are no longer possible. Instead, the space must be divided into hypercuboids with unequal side lengths to ensure that they become $[0,1]^{m}$ under conjugation. We must also ensure that these hypercuboids are disjoint, since we rely on the small measure of their union to conclude that the number of hypercuboids needed for covering is small. These two restrictions together are difficult to satisfy. This issue becomes even more challenging when dealing with fractals. Overcoming this obstacle is the second main contribution of this paper, discussed in Section 6.

2. Notation

The following notation will be used throughout the paper.

2.1. Homogeneous spaces

Fix $m\in\mathbb{N}$ and set $G=\text{SL}_{m+1}(\mathbb{R}),\Gamma=\text{SL}_{m+1}(\mathbb{Z})$ and $\mathcal{X}=G/\Gamma$ . The space $\mathcal{X}$ can be naturally identified with the space of unimodular lattices in $\mathbb{R}^{m+1}$ , via the identification $A\Gamma\mapsto A\mathbb{Z}^{d}$ .

Fix $w=(w_{1},\ldots,w_{m})\in\mathbb{R}^{m}$ such that $w_{1}\geq w_{2}\geq\ldots\geq w_{m}>0$ and $w_{1}+\ldots+w_{m}=1$ . Define $w_{m+1}:=w_{1}$ and set

\displaystyle w_{I}=\sum_{i\in I}w_{i}

(5)

for any subset $I$ of $\{1,\ldots m+1\}$ . For $t>0$ , define

\displaystyle g_{t}=\begin{pmatrix}t^{w_{1}}\\ &\ddots\\ &&t^{w_{m}}\\ &&&t^{-1}\end{pmatrix},\quad a_{t}=\begin{pmatrix}e^{tw_{1}}\\ &\ddots\\ &&e^{tw_{m+1}}\end{pmatrix}.

(6)

As in the Introduction, we define for $\theta\in\mathbb{R}^{m}$ , the matrix

\displaystyle u(\theta)=\begin{pmatrix}I_{m}&\theta\\ &1\end{pmatrix}.

(7)

We define $H$ to be the following subgroup of $G$ ,

\displaystyle H=\left\{\begin{pmatrix}t_{1}&&&x_{1}\\ &\ddots&&\vdots\\ &&t_{m}&x_{m}\\ &&&(t_{1}\ldots t_{m})^{-1}\end{pmatrix}:(t_{1},\ldots,t_{m})\in\mathbb{R}_{+}^{m},(x_{1},\ldots,x_{m})\in\mathbb{R}^{m}\right\}.

(8)

2.2. Iterated Function Systems

A contracting similarity is a map $\mathbb{R}\rightarrow\mathbb{R}$ of the form $x\mapsto cx+y$ where $c\in(0,1)$ , and $y\in\mathbb{R}^{d}$ . A finite similarity Iterated Function System with constant ratio (IFS) on $\mathbb{R}$ is a collection of contracting similarities $\Phi=(\phi_{e}:\mathbb{R}\rightarrow\mathbb{R})_{e\in E}$ indexed by a finite set $E$ , called the alphabet, such that there exists a constant $c\in(0,1)$ independent of $e$ so that

\phi_{e}(x)=cx+w_{e},

for all $e\in E$ .

Let $B=E^{\mathbb{N}}$ . The coding map of an IFS $\Phi$ is the map $\tau:B\rightarrow\mathbb{R}$ defined by the formula

\displaystyle\tau(b)=\lim_{l\rightarrow\infty}\phi_{b_{1}}\circ\ldots\circ\phi_{b_{l}}(0).

(9)

It is well known that the limit in $\eqref{eq:def tau}$ exists and that the coding map is continuous. The image of $B$ under the coding map called the limit set of $\Phi$ , is a compact subset of $\mathbb{R}$ , which we denote by $\mathcal{K}=\mathcal{K}(\Phi)$ . We define for ${\tilde{e}}=(e_{1},\ldots,e_{l})\in E^{p}$ ,

\displaystyle\mathcal{K}_{\tilde{e}}=\phi_{e_{1}}\circ\ldots\circ\phi_{e_{l}}(\mathcal{K}),\quad\mathcal{F}(l)=\{\mathcal{K}_{{\tilde{e}}^{\prime}}:{\tilde{e}}^{\prime}\in E^{l}\}.

(10)

We will say that $\Phi$ satisfies an open set condition (OSC for short) if there exist a non-empty open subset $U\subset\mathbb{R}$ such that the following holds

	$\displaystyle\phi_{e}(U)\subset U$	$\displaystyle\text{ for every }e\in E$
	$\displaystyle\phi_{e}(U)\cap\phi_{e^{\prime}}(U)=\emptyset,$	$\displaystyle\text{ for every }e\neq e^{\prime}\in E.$

Let $\text{Prob}(E)$ denote the space of probability measures on $E$ . For each $\nu\in\text{Prob}(E)$ we can consider the measure $\eta_{*}\nu^{\otimes\mathbb{N}}$ under the coding map. A measure of the form $\eta_{*}\nu^{\otimes\mathbb{N}}$ is called a Bernoulli measure.

The following proposition is well known (see for eg [[Hutchinson], Thm. 5.3(1)] for a proof).

Proposition 2.1.

Suppose $\Phi=\{\phi_{e}:e\in E\}$ is an IFS satisfying the open set condition with limit set $\mathcal{K}$ . Let $c$ denote the common contraction ratio of $(\phi_{e})_{e\in E}$ and $p=\#E$ . Then the Hausdorff dimension $s$ of $\mathcal{K}$ is $-\log p/\log c$ . Moreover the $s$ -dimensional Hausdorff measure $H^{s}$ satisfies $0<H^{s}(\mathcal{K})<1$ . If $\mu$ denotes the normalised restriction of $H^{s}$ to $\mathcal{K}$ , then $\mu$ is a Bernoulli measure and equals $\eta_{*}\nu^{\otimes\mathbb{N}}$ , where $\nu$ is the uniform measure on $E$ , i.e, $\nu(F)=\#F/\#E$ for all $F\subset E$ . Moreover, there exists a constant $\beta>0$ such that for all $x\in\mathbb{R}$ , we have

\displaystyle\mu([x-r,x+r])\leq\beta r^{s}.

(11)

For the rest of the paper, we fix for $1\leq i\leq m$ , the IFS $\Phi_{i}=\{\phi_{i,e}:e\in E_{i}\}$ with common contraction ratio $c_{i}$ and the limit set $\mathcal{K}_{i}$ . Let $p_{i}=\#E_{i}$ , $s_{i}=-\log p_{i}/\log c_{i}$ , $\mathcal{K}=\prod_{i}\mathcal{K}_{i}$ and $s=\sum_{i}s_{i}$ . Let $\mu_{i}$ denote the normalised restriction of $H^{s_{i}}$ to $\mathcal{K}_{i}$ and define the measure $\mu=\otimes_{i}\mu_{i}$ on $\mathcal{K}$ . Let us define the constant $\eta$ as

\displaystyle\eta

\displaystyle=\min\{s_{i}:i\in\{1,\ldots,m\}\}.

(12)

An observations that will be needed later in the proof is that

\displaystyle\eta\geq\sum_{i\notin I}s_{i}w_{I\cup\{i\}},

(13)

for all $I\subsetneq\{1,\ldots,m\}$ .

3. Weighted Mother Inequality

Let us define

V_{l}=\bigwedge^{l}\mathbb{R}^{m+1},\qquad V=\bigoplus_{l=1}^{m+1}V_{l}.

Define action of $G$ on $V$ via the map $g\mapsto\bigoplus_{l=1}^{m+1}\bigwedge^{l}g$ . Suppose $\{e_{1},\ldots,e_{m+1}\}$ denote the standard basis of $\mathbb{R}^{m+1}$ . For each index set $I=\{i_{1}<\cdots<i_{l}\}\subset\{1,\dots,m+1\}$ , we define

\displaystyle{e}_{I}:={e}_{i_{1}}\wedge\cdots\wedge\mathbf{e}_{i_{l}}.

(14)

The collection of monomials $\mathbf{e}_{I}$ gives a basis of $V_{l}=\bigwedge^{l}\mathbb{R}^{m+1}$ for each $1\leq l\leq m+1$ . For $v\in V$ and each index set $I$ , we denote by $v_{I}\in\mathbb{R}$ , the unique value so that $v=\sum_{J}v_{J}e_{J}$ , where the sum is taken over all index sets $J$ .

For each $l$ , we define quasi-norm $\|.\|$ on each of $V_{l}$ as

\displaystyle\|v\|=\max_{I}|v_{I}|^{\frac{1}{w_{I}}},

(15)

where maximum is taken oven all index sets $I$ of cardinality $l$ and $w_{I}$ is defined as in (5). Note that for all $t\in\mathbb{R}$ and for all $v\in V_{l}$ ( $1\leq l\leq m+1$ ), we have

\displaystyle\|a_{t}v\|=e^{t}\|v\|.

(16)

Definition 3.1.

For a discrete subgroup $\Lambda$ of $\mathbb{R}^{m+1}$ of rank $l\geq 1$ , we define $v_{\Lambda}\in V_{l}/\{\pm 1\}$ as $v_{1}\wedge\ldots\wedge v_{l}$ , where $v_{1},\ldots,v_{l}$ is a $\mathbb{Z}$ -basis of $\Lambda$ . Note that the definition of $v_{\Lambda}$ is independent of the choice of basis $v_{1},\ldots,v_{l}$ . We define $\|\Lambda\|$ as

\displaystyle\|\Lambda\|=\|v\|,

(17)

where $\|.\|$ on $V_{l}$ is defined as in (15). We also define $\|\{0\}\|=1$ .

The main result of this section is the following inequality which originated in [EMM]. Part of this proof is motivated from [[BQ12], Proof of Prop. 3.1] where it is termed the ‘Mother inequality’.

Theorem 3.2.

There exist a constant $D>0$ and a finite set $\{0\}\subset S\subset[0,1)$ such that the following holds. Fix a lattice $\Lambda$ in $\mathbb{R}^{m+1}$ . Then for any sublattices $\Lambda_{1},\Lambda_{2}$ of $\Lambda$ , we have

\displaystyle\min\{\|\Lambda_{1}\cap\Lambda_{2}\|^{a}\|\Lambda_{1}+\Lambda_{2}\|^{1-a}:a\in S\}\leq D\max\{\|\Lambda_{1}\|^{a}\|\Lambda_{2}\|^{1-a}:0<a<1\},

(18)

where $\Lambda_{1}+\Lambda_{2}$ denotes the smallest discrete subgroup of $\mathbb{R}^{m+1}$ containing $\Lambda_{1}$ and $\Lambda_{2}$ .

Proof.

The proof is divided into two cases:

Case $1$ : $\Lambda_{1}\cap\Lambda_{2}=\{0\}$ . In this case, (18) follows if the following holds: For any $p\in V_{i}$ and $q\in V_{j}$ , the following holds:

\displaystyle\|p\wedge q\|\leq\left(\frac{(i+j)!}{i!j!}\right)^{m}\max\{\|p\|^{a}\|q\|^{1-a}:0<a<1\}.

(19)

To prove (19), note that

$\displaystyle\\|p\wedge q\\|$	$\displaystyle=\max_{\#I=i+j}\|(p\wedge q)_{I}\|^{\frac{1}{w_{I}}}$	(20)
	$\displaystyle\leq\max_{\#I=i+j}\left(\sum_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}\|p_{J}\|\|q_{K}\|\right)^{\frac{1}{w_{I}}}$	(21)
	$\displaystyle\leq\left(\frac{(i+j)!}{i!j!}\right)^{m}\max_{\#I=i+j}\left(\max_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}\|p_{J}\|\|q_{K}\|\right)^{\frac{1}{w_{I}}}$	(22)
	$\displaystyle=\left(\frac{(i+j)!}{i!j!}\right)^{m}\max_{\#I=i+j}\max_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}(\|p_{J}\|^{\frac{1}{w_{J}}})^{\frac{w_{J}}{w_{J}+w_{K}}}(\|w_{K}\|^{\frac{1}{w_{K}}})^{\frac{w_{K}}{w_{J}+w_{K}}}\quad\text{using fact that }w_{I}=w_{J}+w_{K}$	(23)
	$\displaystyle\leq\left(\frac{(i+j)!}{i!j!}\right)^{m}\max_{\#I=i+j}\max_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}(\\|p\\|)^{\frac{w_{J}}{w_{J}+w_{K}}}\\|q\\|^{\frac{w_{K}}{w_{J}+w_{K}}}$	(24)
	$\displaystyle\leq\left(\frac{(i+j)!}{i!j!}\right)^{m}\max\{\\|p\\|^{a}\\|q\\|^{1-a}:0<a<1\}.$	(25)

Hence (19) follows.

Case $2$ : $\Lambda_{1}\cap\Lambda_{2}\neq\{0\}$ . In this case, (18) follows if the following holds: For any $p=p_{1}\wedge\ldots\wedge p_{i}\in V_{i}$ , $q=q_{1}\wedge\ldots\wedge q_{j}\in V_{j}$ and $r=r_{1}\wedge\ldots\wedge r_{k}\in V_{k}$ , the following holds:

\displaystyle\min\{\|p\|^{a}\|p\wedge q\wedge r\|^{1-a}:a\in S\}\leq C_{i,j,k}\{\|p\wedge q\|^{a}\|p\wedge r\|^{1-a}:0<a<1\}.

(26)

To prove this, we define the linear map $\psi:V_{i+j}\otimes V_{i+k}\rightarrow V_{i}\otimes V_{i+j+k}$ such that for any $x=x_{1}\wedge\ldots\wedge x_{i+j}$ and $y=y_{1}\wedge\ldots\wedge y_{i+l}$ ,

\psi(x\otimes y)=\sum_{\#I=i}\varepsilon_{I}x_{I}\otimes x_{I^{c}}\wedge y,

where the sum is taken over all subsets $I\subset\{1,\ldots,i+j\}$ of size $i$ . Let us explain each term in the sum.

•

the element $x_{I}$ is the exterior product $x_{I}=x_{l_{1}}\wedge\ldots\wedge x_{l_{r}}$ , when one writes $I=\{l_{1},...,l_{r}\}$ with $i_{1}<\textperiodcentered\textperiodcentered\textperiodcentered<i_{r}$ .
•

the element $x_{I^{c}}$ is the exterior product $x_{I^{c}}=x_{l_{i+1}}\wedge\ldots\wedge x_{l_{i+j}}$ , when one writes $I=\{l_{i+1},...,l_{i+j}\}$ with $l_{i+1}<\textperiodcentered\textperiodcentered\textperiodcentered<l_{i+j}$ .
•

the sign $\varepsilon_{I}$ is the signature of the permutation of $\{1,...,i+j\}$ sending $z$ to $l_{z}$ for $1\leq p\leq i+j$ .

For $1\leq l,l^{\prime}\leq m+1$ , we define quasi-norms $\|.\|$ on $V_{l}\otimes V_{l^{\prime}}$ as

\displaystyle\|x\|=\max_{\#I=l,\#J=l^{\prime}}|x_{I,J}|^{\frac{1}{w_{I}+w_{J}}},

(27)

where $x=\sum_{\#I=l,\#J=l^{\prime}}x_{I,J}e_{I}\otimes e_{J}$ . Let

S_{l,l^{\prime}}=\{\frac{w_{I}}{w_{I}+w_{J}}:\#I=l,\#J=l^{\prime}\}.

Then it is easy to see that for every $x\in V_{l}$ and $y\in V_{l^{\prime}}$ , we have $a\in S_{l,l^{\prime}}$ such that

\displaystyle\|p\wedge q\|=\|p\|^{a}\|q\|^{1-a}.

(28)

Let $S^{\prime}=\cup_{l,l^{\prime}}S_{l,l^{\prime}}$ and $S=S^{\prime}\cup\{0\}$ . It is clear that $S$ is a finite set, not containing $1$ . Also note that $0\notin S^{\prime}$ .

Note that for $1\leq l,l^{\prime}\leq m+1$ , the action of $a_{t}$ (defined as in (6)) on $V_{l}\otimes V_{l^{\prime}}$ via the map $\wedge^{l}a_{t}\otimes\wedge^{l^{\prime}}a_{t}$ satisfies

\displaystyle\|a_{t}w\|=e^{t}\|w\|.

(29)

Also for all $x\in V_{i+j}$ , $y\in V_{i+k}$ , we have

\displaystyle\psi(a_{t}(x\otimes y))=a_{t}\psi(x\otimes y).

(30)

We define

\displaystyle C_{i,j,k}=\sup\{\|\psi(x)\|:x\in V_{i+j}\otimes V_{i+k},\|x\|=1\},

(31)

which is a finite value since $\psi$ is continuous and the set $\{x\in V_{i+j}\otimes V_{i+k}:\|x\|=1\}$ is a compact set.

Fix $p=p_{1}\wedge\ldots\wedge p_{i}\in V_{i}$ , $q=q_{1}\wedge\ldots\wedge q_{j}\in V_{j}$ and $r=r_{1}\wedge\ldots\wedge r_{k}\in V_{k}$ . Note that (26) hold trivially if $\|p\wedge q\|=0$ or $\|p\wedge r\|=0$ . Hence, we may assume $\|p\wedge q\otimes p\wedge r\|\neq 0$ and let $s=\log\|p\wedge q\otimes p\wedge r\|$ . Then, we have using (29) that

\displaystyle\|a_{-s}(p\wedge q\otimes p\wedge r)\|=1.

(32)

Note that

\displaystyle\psi(p\wedge q\otimes p\wedge r)=p\otimes q\wedge p\wedge r.

(33)

Hence, we have

$\displaystyle\min\{\\|p\\|^{a}\\|p\wedge q\wedge r\\|^{1-a}:a\in S\}$	$\displaystyle\leq\\|p\otimes p\wedge q\wedge r\\|$	using (28)
	$\displaystyle=\\|p\otimes q\wedge p\wedge r\\|$
	$\displaystyle=\\|\psi(p\wedge q\otimes p\wedge r)\\|$	using (33)
	$\displaystyle=e^{s}\\|a_{-s}\psi(p\wedge q\otimes p\wedge r)\\|$	using (29)
	$\displaystyle=e^{s}\\|\psi(a_{-s}(p\wedge q\otimes p\wedge r))\\|$	using (30)
	$\displaystyle\leq e^{s}C_{i,j,k}$	using (31), (32)
	$\displaystyle=e^{s}C_{i,j,k}\\|a_{-s}(p\wedge q\otimes p\wedge r)\\|$	using (32)
	$\displaystyle=C_{i,j,k}\\|p\wedge q\otimes p\wedge r\\|$	using (29)
	$\displaystyle\leq C_{i,j,k}\max\{\\|p\\|^{a}\\|q\\|^{1-a}:0<a<1\}$	$\displaystyle\text{using \eqref{eq: w 2}}.$

Thus theorem now follows by taking any $D$ larger than all the constants appearing above. ∎

4. Critical Exponent

For $g\in G$ , we define

\displaystyle\|g\|_{V_{l}}:=\sup\left\{\|gv\|:v\in V_{l},\|v\|=1\right\}.

(34)

Also, for any compact subset $Q\subset H$ (defined as in (8)), define the set $\tilde{Q}=\{a_{s}ha_{-s}:s>0,h\in Q\}$ . It is easy to see that $\tilde{Q}$ is still a compact set. Define

\displaystyle\|{Q}\|=\sup\{\|gv\|_{V_{l}},\|g^{-1}v\|_{V_{l}}:1\leq l\leq m,g\in\tilde{Q},v\in V_{l},\|v_{l}\|=1\},

(35)

Lemma 4.1.

Fix $1\leq l\leq m$ . Then for all $g\in H$ , $v\in V_{l}$ with $\|v\|\leq 1$ , we have

\|gv\|\leq\|\{g\}\|\|v\|

Proof.

Let $t=-\log\|v\|>0$ . Then by (16), we have $\|a_{t}v\|=1$ . Thus we have

	$\displaystyle\\|gv\\|$	$\displaystyle=e^{-t}\\|a_{t}ga_{-t}a_{t}v\\|\quad\text{using \eqref{eq: \|\|\|\| is a t invariant}}$
		$\displaystyle\leq e^{-t}\\|a_{t}ga_{-t}\\|_{V_{l}}$
		$\displaystyle\leq\\|v\\|\\|\{g\}\\|.$

∎

Recall the following well-known fact.

Lemma 4.2.

There exists $c\geq 1$ such that the following holds for all $1\leq l\leq m$ and all $v,w\in V_{l}$

\|v+w\|\leq c(\|v\|+\|w\|)

The following lemma is a simple application of Fubini’s Theorem.

Lemma 4.3.

Let $\nu$ be a Borel measure and $f$ a non-negative Borel function on a separable metric space $X$ . Then,

\int_{X}f\;d\nu=\int_{0}^{\infty}\nu\left(\{x\in X:f(x)\geq R\}\right)\;dR

The following Proposition studies the average rate of expansion of vectors in the exterior powers of $\mathbb{R}^{m+1}$ , and is the main result in this section.

Proposition 4.4.

Fix $1\leq l\leq m$ . For all $0<\rho<1$ , there exists $C=C(\rho)>1$ such that the following holds for all $v\in V_{l}\setminus\{0\}$ ,

\int_{\mathcal{K}}\|g_{t}u(x)v\|^{-\eta\rho}\,d\mu(x)\leq Ct^{-\eta\rho}\|v\|^{-\eta\rho}.

Proof.

Fix $1\leq l\leq m$ , $0<\rho<1$ and $v\in V_{l}$ . Without loss of generality, we may assume that $\|v\|=1$ . Consider the vector space $V_{l}^{+}\subset V_{l}$ spanned by $\{e_{I}:I\subset\{1,\ldots,d\}\}$ . Also consider the vector space $V_{l}^{-}$ spanned by $\{e_{I}:I\nsubseteq\{1,\ldots,d\}\}$ We will denote by $\pi_{+}:V_{l}=V_{l}^{+}\oplus V_{l}^{-}\rightarrow V_{l}^{+}$ the canonical projection. Similarly define $\pi_{-}:V_{l}\rightarrow V_{l}^{-}$ . Note that for all $v\in V_{l}$ , we have

\|v\|=\max\{\|\pi_{+}(v)\|,\|\pi_{-}(v)\|\}

Then we have

\displaystyle\int_{\mathcal{K}}\|g_{t}u(x)v\|^{-\eta\rho}\,d\mu(x)\leq\int_{\mathcal{K}}\|\pi_{+}(g_{\tau(x)}u(x)v)\|^{-\eta\rho}\,d\mu(x)=\int_{\mathcal{K}}t^{-\eta\rho}\|\pi_{+}(u(x)v)\|^{-\eta\rho}\,d\mu(x).

Thus, it is enough to prove that the quantity $\int_{\mathcal{K}}\|\pi_{+}(u(x)v)\|^{-\eta\rho}\,d\mu(x)$ is bounded above by a constant independent of $v$ .

To do this, we define $K=\|\{u(x):x\in\mathcal{K}\}\|$ , where $\|.\|$ is defined as in (35). Since $\mu$ is compactly supported, we have that $K<\infty$ .

Case 1 $\|\pi_{-}(v)\|<1/3Kc)$ . Then, since $\|v\|=1$ , we have $\|\pi_{+}(v)\|>2/3$ . This means that

	$\displaystyle\\|\pi_{+}(u(x)v)\\|$	$\displaystyle\geq c^{-1}\\|\pi_{+}(u(x)\pi_{+}(v))\\|-\\|\pi_{+}(u(x)\pi_{-}(v))\\|\quad\text{using Lemma \ref{lem: triangle inequality}}$
		$\displaystyle\geq c^{-1}\\|\pi_{+}(v)\\|-\\|u(x)\pi_{-}(v)\\|$
		$\displaystyle\geq\frac{2}{3c}-K.\frac{1}{3Kc}\quad\text{using Lemma \ref{lem: operator norm make sense} and fact that $\\|\pi_{-}(v)\\|\leq 1$}$
		$\displaystyle=\frac{1}{3c}.$

Thus, $\int_{\mathcal{K}}\|\pi_{+}(u(x)v)\|^{-\eta\rho}\,d\mu(x)\leq(3c)^{-\eta\rho}$ .

Case 2 $\|\pi_{-}(v)\|>1/3Kc$ . In this case, there exists a subset $\{m+1\}\subset I\subset\{1,\ldots,m+1\}$ of cardinality $l$ such that $v_{I}>1/K^{\prime}$ , where $K^{\prime}=\max\{(3cK)^{w_{I}}:\{m+1\}\subset I\subset\{1,\ldots,m+1\},\#I=l\}$ . Let us define for all $i\notin I$ , the set $I_{i}=\{I\cup\{i\}\setminus\{m+1\}\}$

Let us define

	$\displaystyle E(v,R)$	$\displaystyle=\{x\in\mathbb{R}^{m}:\\|\pi_{+}(u(x)v)\\|<R\},$
	$\displaystyle E^{\prime}(v,R)$	$\displaystyle=\{x\in\mathbb{R}^{m}:\text{for all $i\notin I$, we have }(\pi_{+}(u(x)v))_{I_{i}}<R\}.$

It is clear that $E(v,R)\subset E^{\prime}(v,R)$ . Claim that for all $R<1$ , we have

\displaystyle\mu(E^{\prime}(v,R))\leq\beta^{m+1-l}(K^{\prime})^{\sum_{i\not inI}s_{i}}R^{\eta},

(36)

where $\beta$ is large enough so that (11) holds for all $\mu_{i}$ ( $1\leq i\leq m$ ). To see this, first of all, note that by explicit computation, we have for every $J\subset\{1,\ldots,m\}$ , the following holds

(\pi_{+}(u(x)v))_{J}=v_{J}+\sum_{j\in J}e_{j,J}v_{J\cup\{m+1\}\setminus\{j\}}x_{j},

for some $e_{j,J}\in\{\pm 1\}$ . Thus for all $i\notin I$ , we have

(\pi_{+}(u(x)v))_{I_{i}}=e_{I_{i},i}v_{I}x_{i}+\left(v_{I_{i}}+\sum_{j\in I\setminus\{m+1\}}e_{j,I_{i}}v_{I_{i}\cup\{m+1\}\setminus\{j\}}x_{j}\right),

Thus if we fix $x_{j}$ for $j\in I\setminus\{m+1\}$ , then for all $i\notin I$ and $R>0$ , we have that the condition $|(\pi_{+}(u(x)v))_{I_{i}}|^{1/{w_{I_{i}}}}<R$ is equivalent to the condition that $x_{i}$ belongs to an interval of size atmost $R^{w_{I_{i}}}/|v_{I}|<K^{\prime}R^{w_{I_{i}}}$ , which has $\mu_{i}$ measure less than $\beta(K^{\prime}R^{w_{I_{i}}})^{s_{i}}$ (using (11)). Hence, by Fubini’s Theorem, we get that

\displaystyle\mu(E^{\prime}(v,R))\leq\beta^{m+1-l}(K^{\prime})^{\sum_{i\notin I}s_{i}}R^{\sum_{i\notin I}s_{i}w_{I_{i}}}.

(37)

From (13), we know that $\sum_{i\notin I}s_{i}w_{I_{i}}\geq\eta$ . Coming this with the fact that $R<1$ , we get (36) follows from (37).

Now using Lemma 4.3, we get that

	$\displaystyle\int_{\mathcal{K}}\\|\pi_{+}(u(x)v)\\|^{-\eta\rho}\,d\mu(x)$	$\displaystyle=\int_{0}^{\infty}\mu(E(v,R^{-1/\eta\rho}))\,dR$
		$\displaystyle\leq\int_{0}^{1}\mu(E(v,R^{-1/\eta\rho}))\,dR+\int_{1}^{\infty}\mu(E(v,R^{-1/\eta\rho}))\,dR$
		$\displaystyle\leq 1+\beta^{m+1-l}(K^{\prime})^{\sum_{i\not inI}s_{i}}\int_{1}^{\infty}R^{-1/\rho}\,dR=:C^{\prime}<\infty,$

where in last inequality we used fact that $\mu$ is a probability measure and (36). Thus the proposition holds by taking $C=\max\{(3c)^{-eta\rho},C^{\prime}\}$ . ∎

5. Height Functions

We will need some notation before proceeding. For $\Lambda\in\mathcal{X}$ , let $P(\Lambda)$ denote the set of all primitive subgroups of the lattice $\Lambda$ , i.e, the subgroups $L$ of the lattice $\Lambda$ satisfying $L=\Lambda\cap\mathrm{span}_{\mathbb{R}}(L)$ , where $\mathrm{span}_{\mathbb{R}}(L)$ is the smallest vector subspace containing $L$ . For every $0<l\leq m+1$ , we define $\varphi_{l}:X\rightarrow[1,\infty)$ as

\displaystyle\varphi_{l}(x)=\max\{\|L\|^{-1}:L\in P(x),\text{ rank}(L)=l\},

(38)

where $\|.\|$ is defined as in (17). For $l=0$ , we define $\varphi_{l}\equiv 1$ . It is easy to see that $\varphi_{m+1}\equiv 1$ .

Also, let $D>0$ and $S\subset[0,1)$ be defined as in Theorem 3.2.

Lemma 5.1.

For every $\Lambda\in\mathcal{X}$ and $1\leq l\leq m$ , we have $\varphi_{l}(\Lambda)\geq 1$ .

Proof.

Fix $1\leq l\leq m$ and $\Lambda\in\mathcal{X}$ . It is easy to see that there exists a primitive sublattice $\Lambda_{l}$ of rank $l$ of co-volume less than or equal to $1$ . This follows from a simple application of induction and Minkowski’s Convex Body Theorem. This means that the vector $v_{\Lambda_{l}}=\sum_{I}v_{\Lambda_{l},I}e_{I}\in V_{l}$ (defined as in Def. 3.1) satisfies $\sqrt{\sum_{I}|v_{\Lambda_{l},I}|}\leq 1$ , where the sum is taken over all index sets $I$ of cardinality $l$ . Hence we have $|v_{\Lambda_{l},I}|\leq 1$ , which gives $\|\Lambda_{l}\|\leq 1$ . Thus, $\varphi_{l}(\Lambda)\geq\|\Lambda_{l}\|^{-1}\geq 1$ .

∎

Lemma 5.2.

For any compact subset $Q\subset H$ , the following holds for every $h\in Q$ , $\Lambda\in{\mathcal{X}}$ ,

\displaystyle\|Q\|^{-1}{\varphi}_{l}(\Lambda)\leq{\varphi}_{l}(h\Lambda)\leq\|Q\|{\varphi}_{l}(\Lambda),

(39)

for all $1\leq l\leq m$ .

Proof.

Using Lemma 5.1, we have that there exists a $\Lambda_{l}\in P(\Lambda)$ of rank $l$ such that $\|\Lambda_{l}\|\leq 1$ and $\varphi_{l}(\Lambda)=\|\Lambda_{l}\|^{-1}$ . Let $v_{\Lambda_{l}}$ be defined as in Def. 3.1. Then we have $\|v_{\Lambda_{l}}\|\leq 1$ . Hence we get

\displaystyle\varphi_{l}(\Lambda)

\displaystyle=\frac{1}{\|v_{\Lambda_{l}}\|}\leq\frac{\|Q\|}{\|hv_{\Lambda_{l}}\|}\leq\|Q\|\varphi_{l}(h\Lambda),

where penultimate inequality follows by Lemma 4.1. Similarly, we have

\varphi(h\Lambda)\leq\|Q\|\varphi(h^{-1}h\Lambda)=\|Q\|\varphi(\Lambda).

Thus the lemma holds. ∎

The following Proposition introduces the method of ‘integral inequalities’ into the proof and is the place where weighted Mother inequality is used.

Proposition 5.3.

For every $0<\rho<1$ , there exists a constant $C=C(\rho)$ , such that for every $t\in\mathbb{R}$ and $1\leq l\leq m$ ., there exists $\zeta=\zeta(t)\geq 1$ , so that for all $\Lambda\in{\mathcal{X}}$ , we have

	$\displaystyle\int_{\mathcal{K}}{\varphi}_{l}^{\eta\rho}(g_{t}u(x)\Lambda)\,d\mu({x})$	$\displaystyle\leq Ct^{-\eta\rho}{\varphi}_{l}^{\eta\rho}(\Lambda)$		(40)
		$\displaystyle+\left(\zeta\max\{\varphi_{l-j}^{a}\varphi_{l+j}^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}\right)^{\eta\rho}.$		(41)

Proof.

Fix $1\leq l\leq m$ , $0<\rho<1$ and $t\in\mathbb{R}$ . Let us define

\zeta^{\prime}=\|\{g_{t}u(x):x\in\mathcal{K}\}\|,

and define $\zeta=\max\{D.\zeta^{\prime},1\}$ . Also, define $\Lambda_{l}\in P(\Lambda)$ be sublattice of rank $l$ such that $\varphi_{l}(\Lambda)=\|\Lambda_{l}\|^{-1}$ . Let $v_{\Lambda_{l}}$ be defined as in Def. 3.1. Claim that for all $x\in\mathcal{K}$ , we have

\displaystyle\varphi_{l}(g_{t}u(x)\Lambda)\leq\max\left\{\frac{1}{\|g_{t}u(x)\Lambda_{l}\|},\zeta\max\{\varphi_{l-j}^{a}\varphi_{l+j}^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}\right\}.

(42)

To see this, note that if $\varphi_{l}(g_{t}u(x)\Lambda)\neq\|g_{t}u(x)\Lambda_{l}\|^{-1}$ , then the following holds. Using Lemma 5.1, we get that in this case, there exists $\Lambda_{l,x}\in P(\Lambda)$ with $\Lambda_{l,x}\neq\Lambda_{l}$ and $\|g_{t}u(x)\Lambda_{l,x}\|\leq 1$ such that

	$\displaystyle\varphi_{l}(g_{t}u(x)\Lambda)$	$\displaystyle=\\|g_{t}u(x)\Lambda_{l,x}\\|^{-1}$
		$\displaystyle\leq\frac{\zeta^{\prime}}{\\|(g_{t}u(x))^{-1}g_{t}u(x)\Lambda_{l,x}\\|}\quad\text{using Lemma \ref{lem: operator norm make sense}}$
		$\displaystyle=\frac{\zeta^{\prime}}{\\|\Lambda_{l,x}\\|}$
		$\displaystyle\leq\zeta^{\prime}\min\{\frac{1}{\\|\Lambda_{l,x}\\|^{a}\\|\Lambda_{l}\\|^{1-a}}:0<a<1\}$
		$\displaystyle\leq D\zeta^{\prime}\max\{\frac{1}{\\|\Lambda_{l}\cap\Lambda_{l,x}\\|^{a}\\|\Lambda_{l}+\Lambda_{l,x}\\|^{1-a}}:a\in S\}\quad\text{using Theorem \ref{thm: Mother Ineq}}$
		$\displaystyle\leq\zeta\max\{\varphi_{l-j}^{a}\varphi_{l+j}^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}$

From (42), we have

	$\displaystyle\int_{\mathcal{K}}{\varphi}_{l}^{\rho\eta}(g_{t}u(x)\Lambda)\,d\mu({x})$
	$\displaystyle\leq\int_{\mathcal{K}}\frac{1}{\\|g_{t}u(x)v_{\Lambda_{l}}\\|^{\eta\rho}}\,d\mu({x})+\left(\zeta\max\{\varphi_{l-j}^{a}\varphi_{l+j}^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}\right)^{\eta\rho}$
	$\displaystyle\leq C(\rho,l)t^{-\rho\eta}\frac{1}{\\|v_{\Lambda_{l}}\\|^{\eta\rho}}+\left(\zeta\max\{\varphi_{l-j}^{a}\varphi_{l+j}^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}\right)^{\eta\rho},$

where the last inequality follows from Proposition 4.4. Hence, the proposition follows with $C(\rho)=\max_{l}C(\rho,l)$ . ∎

Lemma 5.4.

There exists constants $\alpha_{0}>\alpha_{1}>\ldots>\alpha_{m+1}$ such that

\displaystyle\alpha_{i}\geq 1+\max\{a\alpha_{i-j}+(1-a)\alpha_{i+j}:a\in S\},

(43)

for all $1\leq i\leq m+1$ and $1\leq j\leq\min\{i,m+1-i\}$ .

Proof.

Let $c=\sup S<1$ . We will define $\alpha_{l}$ by induction. Let $\alpha_{0}=1$ and suppose that $\alpha_{k}$ is already defined for $0\leq k<l$ satisfying $\alpha_{0}>\alpha_{1}>\ldots>\alpha_{l-1}$ and (43) for $i,j$ satisfying $i+j<l$ . Then define $\alpha_{l}$ inductively as

\alpha_{l}=-\frac{1}{1-c}+\min\{\alpha_{l-1},\frac{1}{1-c}\min_{1\leq j\leq l/2}\{\alpha_{l-j}-c\alpha_{l-2j}\}\}.

Then, it is clear that $\alpha_{l}<\alpha_{l-1}$ . Also for $1\leq i\leq l$ , $1\leq\min\{i,m+1-i\}j\leq$ such that $i+j\leq l$ , we have

	$\displaystyle\alpha_{i}-\max\{a\alpha_{i-j}+(1-a)\alpha_{i+j}:a\in S\}$
	$\displaystyle\geq\alpha_{i}-(c\alpha_{i-j}+(1-c)\alpha_{i+j})$	using $\alpha_{i-j}<\alpha_{i+j}$ and the definition of $c$
	$\displaystyle=(1-c)\left(\frac{1}{1-c}\left(\alpha_{l-j}-(c\alpha_{l-2j}\right)-\alpha_{l}\right)$
	$\displaystyle\geq 1$	$\displaystyle\text{using the definition of }\alpha_{l}.$

Thus (43) holds for $i,j$ satisfying $i+j\leq l$ . Hence, by induction, the sequence exists. ∎

Fix $\alpha_{0},\ldots,\alpha_{m+1}$ as in Lemma 5.4. For $0<\varepsilon<1$ and $0<\rho<1$ , we define the function

\displaystyle f_{\varepsilon,\rho}(\Lambda)=\sum_{l=0}^{m+1}\varepsilon^{\alpha_{l}}\varphi_{l}^{\eta\rho}(\Lambda)

(44)

Proposition 5.5.

For every $0<\rho<1$ , there exists a constant $C_{\rho}>0$ , depending only on $\rho$ such that for every $t>1$ , there exists $b=b(t)\geq 0$ and $0<\varepsilon=\varepsilon(t)<1$ so that for all $\Lambda\in\mathcal{X}$ , the following holds

\displaystyle\int_{\mathcal{K}}f_{\varepsilon,\rho}(g_{t}u(x)\Lambda)\,d\mu(x)\leq C_{\rho}t^{-\eta\rho}f_{\varepsilon,\rho}(\Lambda)+b.

(45)

Proof.

Fix $t>1$ and $0<\rho<1$ . Let $C,\zeta$ be the constants provided by Proposition 5.3. Let $0<\varepsilon<1$ be a constant to be determined. Suppose $\Lambda\in\mathcal{X}$ . Then using Proposition 5.3, we get that

	$\displaystyle\int_{\mathcal{K}}f_{\varepsilon,\rho}(g_{t}u(x)\Lambda)\,d\mu(x)$
	$\displaystyle=\varepsilon^{\alpha_{0}}+\varepsilon^{\alpha_{m+1}}+\int_{\mathcal{K}}\sum_{l=1}^{m}\varepsilon^{\alpha_{l}}\varphi_{l}^{\eta\rho}(g_{t}u(x)\Lambda)\,d\mu(x)$
	$\displaystyle\leq\varepsilon^{\alpha_{0}}+\varepsilon^{\alpha_{m+1}}+Ct^{-\eta\rho}\sum_{l=1}^{m}\varepsilon^{\alpha_{l}}\varphi_{l}^{\eta\rho}(\Lambda)$
	$\displaystyle+\sum_{l=1}^{m}\varepsilon^{\alpha_{l}}\zeta^{\eta\rho}\max\{(\varphi_{l-j}^{\eta\rho})^{a}(\varphi_{l+j}^{\eta\rho})^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}$		(46)

Note that

	$\displaystyle\varphi_{i-j}^{\eta\rho}(x)$	$\displaystyle\leq\varepsilon^{-\alpha_{i-j}\eta\rho}f_{\varepsilon,\rho}(x),$
	$\displaystyle\varphi_{i+j}^{\eta\rho}(x)$	$\displaystyle\leq\varepsilon^{-\alpha_{i+j}\eta\rho}f_{\varepsilon,\rho}.$

Thus, we have

	$\displaystyle\varepsilon^{\alpha_{i}}\max\{(\varphi_{i-j}^{\eta\rho}(x))^{a}(\varphi_{i+j}^{\eta\rho}(x))^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}$
	$\displaystyle\leq\varepsilon^{\alpha_{i}}\max\{\varepsilon^{-(a\alpha_{i-j}+(1-a)\alpha_{i+j})\eta\rho}f_{\varepsilon,\rho}(x):a\in S,1\leq j\leq\min\{l,m+1-l\}\}$
	$\displaystyle=f_{\varepsilon,\rho}(x)\max\{\varepsilon^{\alpha_{i}-(a\alpha_{i-j}+(1-a)\alpha_{i+j})\eta\rho}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}$
	$\displaystyle\leq\varepsilon f_{\varepsilon,\rho}(x)\quad\text{using \eqref{eq: lem alpha i condition} and fact that $\eta\rho\leq 1$.}$		(47)

Thus, we get from (46) and (47) that

\displaystyle\int_{\mathcal{K}}f_{\varepsilon,\rho}(g_{t}u(x)\Lambda)\,d\mu(x)\leq Ct^{-\eta\rho}f_{\varepsilon,\rho}(\Lambda)+(\varepsilon^{\alpha_{0}}+\varepsilon^{\alpha_{m}})(1-Ct^{-\eta\rho})+m\varepsilon\zeta^{\eta\rho}f_{\varepsilon,\rho}(\Lambda)

Choosing $\varepsilon=\frac{Ct^{-\eta\rho}}{m\zeta}$ and $b=(\varepsilon^{\alpha_{0}}+\varepsilon^{\alpha_{m}})(1-Ct^{-\eta\rho})$ , and $C_{\rho}=2C$ , we get (45). Thus the proposition follows. ∎

6. The Contraction Hypothesis

Definition 6.1 (The Contraction Hypothesis).

Suppose $Y$ is a metric space equipped with an action of $G$ . Given a collection of functions ${f_{t}:Y\rightarrow(0,\infty]:t\in S}$ for some unbounded set $S\subset(0,\infty)$ and real numbers $0<\beta$ , we say that $\mu$ satisfies the $((f_{t})_{t},\beta)$ -contraction hypothesis on $X$ if the following properties hold:

(1)

The set $Z=\{f_{t}=\infty\}$ is independent of $t$ and is $G$ -invariant.
(2)

For every $t\in S$ , $f_{t}$ is uniformly log Lipschitz with respect to the $H$ (defined in (8)) action. That is for every bounded neighborhood $\mathcal{O}$ of identity in $H$ , there exists a constant $C_{\mathcal{O}}\geq 1$ such that for every $g\in\mathcal{O}$ , $y\in Y$ and $t\in\mathbb{N}$ ,

$\displaystyle C_{\mathcal{O}}^{-1}f_{t}(y)\leq f_{t}(gy)\leq C_{\mathcal{O}}f_{t}(y).$ (48)
(3)

There exists a constant $c\geq 1$ such that the following holds: for every $t\in S$ , there exists $T>0$ such that for all $y\in Y$ with $f_{t}(y)>T$ ,

$\displaystyle\int_{\mathcal{K}}f_{t}(g_{t}u(x)y)\,d\mu(x)\leq cf_{t}(y)t^{-\beta}.$ (49)

The functions $f_{t}$ will be referred to as height functions.

Remark 6.2.

The above definition of Contraction Hypothesis is motivated from the corresponding definition in [Khalilsing].

Theorem 6.3.

Let $Y$ be a metric space equipped with an action by $G$ . Assume that $\mu$ satisfies the $(\{f_{t}\}_{t\in S},\beta)$ -contraction hypothesis on $Y$ for a real numbers $0<\beta\leq s$ . Then for all $y\in X\backslash\{f_{t}=\infty\}$ ,

\displaystyle\dim_{H}\left(x\in\mathcal{K}:f_{s}(a_{t}u(x)y)\rightarrow_{t\rightarrow\infty}\infty\text{ for all }s\in S\right)\leq s-\frac{\beta}{1+w_{1}}.

Proof.

Without loss of generality, we may assume that $0\in\mathcal{K}$ . Now fix $\alpha>0$ large enough so that $\mathcal{K}\subset[-\alpha,\alpha]^{m}$ . Let us define $O\subset H$ as

\displaystyle O=\left\{\frac{1}{(t_{1}\ldots t_{m})^{1/(m+1)}}\begin{pmatrix}t_{1}\\ &\ddots\\ &&t_{m}\\ &&&1\end{pmatrix}u(x):x\in[-2\alpha,2\alpha]^{m},c_{i}\leq t_{i}\leq c_{i}^{-1}\text{ for all }1\leq i\leq m\right\}.

(50)

Let $A>1$ be the constant so that (48) holds for all $t\in S$ , $y\in Y$ and $h\in O$ with $C_{O}=A$ .

Fix $y\in Y$ . Fix $t\in S$ large enough so that $t>c_{i}^{-1}$ for all $i$ . For the sake of simplicity, let $g=g_{t}$ . For every $k\in\mathbb{N}$ and $1\leq j\leq m$ , we define $N_{k}(j)$ as the unique integer such that $c_{j}^{N_{k}(j)}\leq t^{-k(1+w_{j})}<c_{j}^{N_{k}(j)-1}$ . We define for all $k\in\mathbb{N}$ , the element $h_{k}\in G$ as

\displaystyle h_{k}=\frac{1}{(c_{1}^{-N_{k}(1)}\ldots c_{m}^{-N_{k}(m)})^{1/(m+1)}}\begin{pmatrix}c_{1}^{-N_{k}(1)}\\ &\ddots\\ &&c_{m}^{-N_{k}(m)}\\ &&&1\end{pmatrix}.

(51)

Let us quickly recall some notation. For $1\leq i\leq m$ , $\mathcal{K}_{i}$ is the limit set of the IFS $\Phi_{i}=\{\phi_{i,e}:e\in E_{i}\}$ , with common contraction ratio $c_{i}$ and $p_{i}=\#E_{i}$ . The dimension of $\mathcal{K}_{i}$ equals $s_{i}=-\log p_{i}/\log c_{i}$ . The set $\mathcal{K}=\prod_{i}\mathcal{K}_{i}$ has dimension $s=\sum_{i}s_{i}$ . The measure $\mu_{i}$ denote the normalised restriction of $H^{s_{i}}$ to $\mathcal{K}_{i}$ and the measure $\mu$ on $\mathcal{K}$ is defined as $\mu=\otimes_{i}\mu_{i}$ . For $1\leq j\leq m$ and $k\in\mathbb{N}$ , we define $\mathcal{F}_{j}(l)$ as in (10) corresponding to $\Phi_{j}=\{\phi_{j,e}:e\in E_{j}\}$ . We also define $\eta_{j}:E_{j}^{\mathbb{N}}\rightarrow\mathbb{R}$ as in (12). We define $\kappa_{j}:\cup_{l}\mathcal{F}_{j}(l)\rightarrow\mathbb{R}$ as $\kappa_{j}(\phi_{j,e_{1}}\circ\ldots\circ\phi_{j,e_{l}}(\mathcal{K}_{j}))=\phi_{j,e_{1}}\circ\ldots\circ\phi_{j,e_{l}}(0)$ . Let us define $\mathcal{F}(k)=\prod_{j}\mathcal{F}_{j}(N_{k}(j))$ . Also define $\kappa:\cup_{k}\mathcal{F}(k)\rightarrow\mathbb{R}^{m}$ as restriction of $\kappa_{1}\times\ldots\kappa_{m}$ . Note that since $0\in\mathcal{K}$ , so $\kappa(R)$ belongs to $R$ for all $R\in\cup_{k}\mathcal{F}(k)$ .

Let $T^{\prime}>0$ and $c>0$ be so that (49) holds for $f_{t}$ and $y\in Y$ with $f_{t}(y)>T^{\prime}$ . Let $T=A^{2}T$ .

Then we have that

\displaystyle\left\{x\in\mathcal{K}:f_{\tau}(a_{s}u(x)y)\rightarrow_{s\rightarrow\infty}\infty\text{ for all }\tau\in S\right\}\subset\bigcup_{N\in\mathbb{N}}Z(N),

(52)

where $Z(N)=\{x\in\mathcal{K}:f_{t}(g^{n}u(x)y)>T\text{ for all }n\geq N\}$ . Fix $N\in\mathbb{N}$ .

Let us make some easy observations before proceeding:

Observation 1 For all $k\in\mathbb{N}$ , we have $h_{k}g^{-k}$ and $g_{k}h_{k}^{-1}$ belongs to $O$ (defined as in (50)).
Explanation: This follows from the definition of $h_{k}$ along with fact that

g^{k}=\frac{1}{(t^{k(1+w_{1})}\ldots t^{k(1+w_{m})})^{1/(m+1)}}\begin{pmatrix}t^{k(1+w_{1})}\\ &\ddots\\ &&t^{k(1+w_{m})}\\ &&&1\end{pmatrix}.

Observation 2 For $M>N$ , if $R\in\mathcal{F}(M)$ is such that $R\cap Z(N)\neq\emptyset$ , then $f_{t}(g^{M}u(x)y)>T/A$ for all $x\in R$ .
Explanation: Suppose $x\in R\cap Z(N)$ . Then as $x\in Z(N)$ and $M>N$ , we have $f_{t}(g^{M}u(x)y)>T$ . Now if $x^{\prime}\in R$ , then the $j$ -th co-ordinate $x^{\prime}-x$ is less than $2c_{j}^{N_{M}(j)}\alpha\leq 2t^{-M(1+w_{j})}\alpha$ . If define $x^{\prime\prime}\in\mathbb{R}^{m}$ as the vector whose $j$ -th entry is $t^{-M(1+w_{j})}$ times the $j$ -th entry of $x^{\prime}-x$ , then $u(x^{\prime\prime})\in O$ . Thus we have

f_{t}(g^{M}u(x^{\prime})y)=f_{t}(g^{M}u(x^{\prime}-x)g^{-M}g^{M}u(x)y)=f_{t}(u(x^{\prime\prime})g^{M}u(x)y)\geq\frac{f_{t}(g^{M}u(x)y)}{A}=\frac{T}{A}.

Observation 3 For any measurable function $\psi:\mathbb{R}^{m}\rightarrow\mathbb{R}_{+}$ , measurable set $X\subset\mathbb{R}^{m}$ and $M\in\mathbb{N}$ , we have

\sum_{\begin{subarray}{c}R\in\mathcal{F}(M)\\ R\cap X\neq\emptyset\end{subarray}}\int_{R}\psi(x)\,d\mu(x)\leq\sum_{\begin{subarray}{c}R\in\mathcal{F}(M-1)\\ R\cap X\neq\emptyset\end{subarray}}\int_{R}\psi(x)\,d\mu(x).

Explanation: Using the fact that each $(\Psi_{j})$ satisfies open set condition, we get that for any $R\in\mathcal{F}(M-1)$ , the following holds

\int_{R}\psi(x)\,d\mu(x)=\sum_{\begin{subarray}{c}R^{\prime}\in\mathcal{F}(M)\\ R^{\prime}\subset R\end{subarray}}\int_{R^{\prime}}\psi(x)\,d\mu(x).

The observation now follows immediately.

Observation 4 For any measurable function $\psi:Y\rightarrow\mathbb{R}_{+}$ , $M\in\mathbb{N}$ and $R\in\mathcal{F}(M)$ , $y\in Y$ , we have

\int_{R}\psi(h_{M}u(x)y)\,d\mu(x)=\mu(R)\int_{\mathcal{K}}\psi(u(x)h_{M}u(\kappa(R))y)\,d\mu(x).

Explanation: Let us define the matrix $h_{M}^{\prime}$ as

\displaystyle h_{M}^{\prime}=\begin{pmatrix}c_{1}^{N_{M}(1)}\\ &\ddots\\ &&c_{m}^{N_{M}(m)}\end{pmatrix}.

(53)

Using the definition of $\mathcal{F}(M)$ and $\kappa$ , it is easy to see that $R=h_{M}^{\prime}\mathcal{K}+\kappa(R)$ . Since $\mu=\otimes_{i}\mu_{i}$ and $\mu_{i}$ is a Bernoulli measure using the Proposition 2.1 corresponding to uniform measure on $E_{i}$ , we get that $\frac{1}{\mu(R)}\mu|_{R}$ equals pushforward of $\mu$ under the map $x\mapsto h_{M}^{\prime}x+\kappa(R)$ . Thus, we have

	$\displaystyle\int_{R}\psi(h_{M}u(x)y)\,d\mu(x)$	$\displaystyle=\mu(R)\int_{\mathcal{K}}\psi(h_{M}u(h_{M}^{\prime}x+\kappa(R))\,d\mu(x)$
		$\displaystyle=\mu(R)\int_{\mathcal{K}}\psi(h_{M}u(h_{M}^{\prime}x)h_{M}^{-1}h_{M}u(\kappa(R))\,d\mu(x)$
		$\displaystyle=\mu(R)\int_{\mathcal{K}}\psi(u((h_{M}^{\prime})^{-1}h_{M}^{\prime}x)h_{M}u(\kappa(R))\,d\mu(x)$
		$\displaystyle=\mu(R)\int_{\mathcal{K}}\psi(u(x)h_{M}u(\kappa(R))y)\,d\mu(x).$

Observation 5 For any measurable function $\psi:Y\rightarrow\mathbb{R}_{+}$ , $M\in\mathbb{N}$ and $R\in\mathcal{F}(M)$ , $y\in Y$ , we have

\mu(R)\psi(h_{M}u(\kappa(R)y)\leq A\int_{R}\psi(h_{M}u(x)y)\,d\mu(x).

Explanation: Note that for all $x\in\mathcal{K}$ , we have

	$\displaystyle\mu(R)\psi(h_{M}u(\kappa(R))y)$	$\displaystyle\leq\mu(R)A\int_{\mathcal{K}}\psi((g^{M}h_{M}^{-1})u(x)h_{M}u(\kappa(R))y)\,d\mu(x)\quad\text{using Observation 1}$
		$\displaystyle=A\int_{R}\psi((g^{M}h_{M}^{-1})h_{M}u(x)y)\,d\mu(x)\quad\text{ using Observation 4 for $y\mapsto\psi((g^{M}h_{M}^{-1})y)$}$
		$\displaystyle=A\int_{R}\psi(g^{M}u(x)y)\,d\mu(x)$

Observation 6 For $M>N$ and $R\in\mathcal{F}(M-1)$ such that $R\cap Z(N)\neq\emptyset$ , we have

\int_{\mathcal{K}}f_{t}(gu(x)h_{M-1}u(\kappa(R))y)\,d\mu(x)\leq t^{-\beta}f_{t}(h_{M-1}u(\kappa(R))y).

Explanation: Since $R\cap Z(N)\neq\emptyset$ , we get from Observation 2 and fact that $\kappa(R)\in R$ that $f_{t}(g^{M-1}u(\kappa(R))y)>T/A$ . Combining this with Observation 1, we get that $f_{t}(g^{M-1}u(\kappa(R))y)>T/A^{2}=T^{\prime}$ . Now above observation follows by using (49).

Note that for any $M>N$ , we have using Observation 2 that

\displaystyle\sum_{\begin{subarray}{c}R\in\mathcal{F}(M)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\mu(R)

\displaystyle\leq\frac{A}{T}\sum_{\begin{subarray}{c}R\in\mathcal{F}(M)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\int_{R}f_{t}(g^{M}u(x)y)\,d\mu(x).

(54)

Also, we have for $L>N$

$\displaystyle\sum_{\begin{subarray}{c}R\in\mathcal{F}(L)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\int_{R}f_{t}(g^{L}u(x)y)\,d\mu(x)$	$\displaystyle\leq\sum_{\begin{subarray}{c}R\in\mathcal{F}(L-1)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\int_{R}f_{t}(g^{L}u(x)y)\,d\mu(x)\text{ using Observation 3}$
	$\displaystyle\leq A\sum_{\begin{subarray}{c}R\in\mathcal{F}(L-1)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\int_{R}f_{t}(gh_{L-1}u(x)y)\,d\mu(x)\text{ using Observation 1}$
	$\displaystyle=A\sum_{\begin{subarray}{c}R\in\mathcal{F}(L-1)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\mu(R)\int_{\mathcal{K}}f_{t}(gu(x)h_{L-1}u(\kappa(R))y)\,d\mu(x)\text{ using Observation 4}$
	$\displaystyle\leq At^{-\beta}\sum_{\begin{subarray}{c}R\in\mathcal{F}(L-1)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\mu(R)f_{t}(h_{L-1}u(\kappa(R))y)\,d\mu(x)\text{ using Observation 6}$
	$\displaystyle\leq A^{2}t^{-\beta}\sum_{\begin{subarray}{c}R\in\mathcal{F}(L-1)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\int_{R}f_{t}(g^{L-1}u(x)y)\,d\mu(x)\text{ using Observation 5}.$	(55)

Using (54) and iteratively using (55) for $L=M,M-1,\ldots,N+1$ , we get that for any $M>N$ the following holds

\displaystyle\sum_{\begin{subarray}{c}R\in\mathcal{F}(M)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\mu(R)

\displaystyle\leq C_{N}A^{2M}t^{-M\beta},

(56)

where $C_{N}=A^{1-2N}t^{N}\int_{\mathcal{K}}f_{t}(g^{N}u(x)y)\,d\mu(x)/T$ . This

For $1\leq j\leq m$ and $l\in\mathbb{N}$ , we define $P_{l}(j)$ as the unique integer satisfying $c_{j}^{P_{l}(j)}\leq t^{-(1+w_{1})l}<c_{j}^{P_{l}(j)-1}$ . Let us try to cover $Z(N)$ by balls of size less than or equal to $t^{-M(1+w_{1})}$ . We do this by selecting sets in $\prod_{j}\mathcal{F}_{j}(P_{M}(j))$ which intersect $Z(N)$ . Note that there are $\prod_{j}p_{j}^{P_{M}(j)}=\prod_{j}c_{j}^{-s_{j}P_{l}(j)}\leq(c_{1}\ldots c_{m})^{-1}t^{s(1+w_{1})M}$ many elements in $\prod_{j}\mathcal{F}_{j}(P_{M}(j))$ , each of which has equal $\mu$ -measure. Thus, we get that

	$\displaystyle\#\{R\in\prod_{j}\mathcal{F}_{j}(P_{M}(j)):R\cap Z(N)\neq\emptyset\}$	$\displaystyle=(\#\prod_{j}\mathcal{F}_{j}(P_{M}(j))).(\sum_{\begin{subarray}{c}R\in\prod_{j}\mathcal{F}_{j}(P_{M}(j))\\ R\cap Z(N)\neq\emptyset\end{subarray}}\mu(R))$
		$\displaystyle\leq(c_{1}\ldots c_{m})^{-1}t^{s(1+w_{1})M}.(\sum_{\begin{subarray}{c}R\in\mathcal{F}(M)\\ R\cap Z(N)\neq\emptyset\end{subarray}}\mu(R))$
		$\displaystyle\leq C_{N}^{\prime}\left(A^{2}t^{-\beta}t^{s(1+w_{1})}\right)^{M},$

where $C_{N}^{\prime}=C_{N}(c_{1}\ldots c_{m})^{-1}$ . Thus for every $t>0$ , we can cover $Z(N)$ by $C_{N}^{\prime}\left(A^{2}t^{-\beta}t^{s(1+w_{1})}\right)^{M}$ -many hypercuboid of diameter less than or equal to $\max\{c_{j}^{P_{M}(j)}\text{diam}(\mathcal{K}_{j})\}\leq 2\alpha.t^{-M(1+w_{1})}$ . Thus,

\displaystyle H^{s-\gamma}(Z(N))\leq\lim_{M\rightarrow\infty}C_{N}^{\prime}\left(A^{2}t^{-\beta}t^{-s(1+w_{1})}\right)^{M}(2\alpha)^{s-\gamma}t^{-M(1+w_{1})(s-\gamma)},

which is finite for all $\gamma>0$ satisfying $A^{2}t^{-\beta}t^{s(1+w_{1})}t^{-(1+w_{1})(s-\gamma)}\leq 1$ , i.e,

\gamma\leq\frac{1}{1+w_{1}}\left(\beta-\frac{2\log A}{\log t}\right).

This gives that

\displaystyle\dim_{H}(Z(N))\leq s-\frac{1}{1+w_{1}}\left(\beta-\frac{2\log A}{\log t}\right).

(57)

Using fact that $\dim_{H}(\cup_{n}I_{n})=\sup_{n}\dim_{H}(I_{n})$ for any countable collection of borel sets $I_{n}$ and (52), (57), we get that

\dim_{H}(\left\{x\in\mathcal{K}:f_{\tau}(a_{s}u(x)y)\rightarrow_{s\rightarrow\infty}\infty\text{ for all }\tau\in S\right\})\leq s-\frac{1}{1+w_{1}}\left(\beta-\frac{2\log A}{\log t}\right).

Since $A$ is independent of $t$ , letting $t\rightarrow\infty$ (which exists as $S$ is unbounded), we get that the theorem holds. Hence proved. ∎

7. Final Proof

Proof of Theorem 1.4.

Let $0<\rho<1$ be given. Using Proposition 5.5, for every $t>1$ , choose $\varepsilon(t)$ and define the collection of height functions

\{f_{t}:=f_{\varepsilon(t),\rho}:t\in\mathbb{R}\}.

Now it is easy to see that the action of $G$ on ${\mathcal{X}}$ satisfies $(\{f_{t}\},\eta)$ -contraction hypothesis with respect to measure $\mu$ . Indeed the first properties of Definition 6.1 follow immediately from the definition of $f_{k}$ . The second condition follows from Lemma 5.2 and the fact that for all $t$ , $f_{t}$ is a linear combination of $\varphi_{l}$ . The third property follows from Proposition 5.5 and for $c=2C_{\rho}$ and $T=bt^{\eta\rho}/C_{\rho}$ corresponding to each $t$ (Note that the value of $b$ also depends on $t$ ).

Thus by Theorem 6.3, we get that for all $\Lambda\in\mathcal{X}$ the following holds

\displaystyle\dim_{H}\left(x\in\mathcal{K}:f_{s}(a_{t}u(x)y)\rightarrow_{t\rightarrow\infty}\infty\text{ for all }s>1\right)\leq s-\frac{\eta\rho}{1+w_{1}}.

Since $\{x\in\mathcal{K}:f_{s}(a_{t}u(x)\Lambda)\rightarrow_{t\rightarrow\infty}\infty\text{ for all }s>1\}=Div(\Lambda,w)$ , we get that

\dim_{H}(Div(\Lambda,w))\leq s-\frac{\eta\rho}{1+w_{1}}.

Since $0<\rho<1$ is arbitrary, the theorem follows. ∎

8. Concluding Remarks

More generally, one could ask for the dimension of weighted singular matrices. Our approach is applicable in this setting. It is also possible to deal with the product of more general fractals rather than the same contraction ratios. The main changes needed in this case involve the contraction hypothesis. These generalizations are ongoing work in progress.

$\displaystyle\\|p\wedge q\\|$	$\displaystyle=\max_{\#I=i+j}\|(p\wedge q)_{I}\|^{\frac{1}{w_{I}}}$	(20)
	$\displaystyle\leq\max_{\#I=i+j}\left(\sum_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}\|p_{J}\|\|q_{K}\|\right)^{\frac{1}{w_{I}}}$	(21)
	$\displaystyle\leq\left(\frac{(i+j)!}{i!j!}\right)^{m}\max_{\#I=i+j}\left(\max_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}\|p_{J}\|\|q_{K}\|\right)^{\frac{1}{w_{I}}}$	(22)
	$\displaystyle=\left(\frac{(i+j)!}{i!j!}\right)^{m}\max_{\#I=i+j}\max_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}(\|p_{J}\|^{\frac{1}{w_{J}}})^{\frac{w_{J}}{w_{J}+w_{K}}}(\|w_{K}\|^{\frac{1}{w_{K}}})^{\frac{w_{K}}{w_{J}+w_{K}}}\quad\text{using fact that }w_{I}=w_{J}+w_{K}$	(23)
	$\displaystyle\leq\left(\frac{(i+j)!}{i!j!}\right)^{m}\max_{\#I=i+j}\max_{\begin{subarray}{c}\#J=i,\#K=j\\ J\cup K=I\end{subarray}}(\\|p\\|)^{\frac{w_{J}}{w_{J}+w_{K}}}\\|q\\|^{\frac{w_{K}}{w_{J}+w_{K}}}$	(24)
	$\displaystyle\leq\left(\frac{(i+j)!}{i!j!}\right)^{m}\max\{\\|p\\|^{a}\\|q\\|^{1-a}:0<a<1\}.$	(25)

	$\displaystyle\\|gv\\|$	$\displaystyle=e^{-t}\\|a_{t}ga_{-t}a_{t}v\\|\quad\text{using \eqref{eq: \|\|\|\| is a t invariant}}$
		$\displaystyle\leq e^{-t}\\|a_{t}ga_{-t}\\|_{V_{l}}$
		$\displaystyle\leq\\|v\\|\\|\{g\}\\|.$

	$\displaystyle\\|\pi_{+}(u(x)v)\\|$	$\displaystyle\geq c^{-1}\\|\pi_{+}(u(x)\pi_{+}(v))\\|-\\|\pi_{+}(u(x)\pi_{-}(v))\\|\quad\text{using Lemma \ref{lem: triangle inequality}}$
		$\displaystyle\geq c^{-1}\\|\pi_{+}(v)\\|-\\|u(x)\pi_{-}(v)\\|$
		$\displaystyle\geq\frac{2}{3c}-K.\frac{1}{3Kc}\quad\text{using Lemma \ref{lem: operator norm make sense} and fact that $\\|\pi_{-}(v)\\|\leq 1$}$
		$\displaystyle=\frac{1}{3c}.$

	$\displaystyle\varphi_{l}(g_{t}u(x)\Lambda)$	$\displaystyle=\\|g_{t}u(x)\Lambda_{l,x}\\|^{-1}$
		$\displaystyle\leq\frac{\zeta^{\prime}}{\\|(g_{t}u(x))^{-1}g_{t}u(x)\Lambda_{l,x}\\|}\quad\text{using Lemma \ref{lem: operator norm make sense}}$
		$\displaystyle=\frac{\zeta^{\prime}}{\\|\Lambda_{l,x}\\|}$
		$\displaystyle\leq\zeta^{\prime}\min\{\frac{1}{\\|\Lambda_{l,x}\\|^{a}\\|\Lambda_{l}\\|^{1-a}}:0<a<1\}$
		$\displaystyle\leq D\zeta^{\prime}\max\{\frac{1}{\\|\Lambda_{l}\cap\Lambda_{l,x}\\|^{a}\\|\Lambda_{l}+\Lambda_{l,x}\\|^{1-a}}:a\in S\}\quad\text{using Theorem \ref{thm: Mother Ineq}}$
		$\displaystyle\leq\zeta\max\{\varphi_{l-j}^{a}\varphi_{l+j}^{1-a}:a\in S,1\leq j\leq\min\{l,m+1-l\}\}$

On the Hausdorff dimension of Weighted Singular Vectors

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

Definition 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

1.1. Outline of proof

2. Notation

2.1. Homogeneous spaces

2.2. Iterated Function Systems

Proposition 2.1.

3. Weighted Mother Inequality

Definition 3.1.

Theorem 3.2.

Proof.

4. Critical Exponent

Lemma 4.1.

Proof.

Lemma 4.2.

Lemma 4.3.

Proposition 4.4.

Proof.

5. Height Functions

Lemma 5.1.

Proof.

Lemma 5.2.

Proof.

Proposition 5.3.

Proof.

Lemma 5.4.

Proof.

Proposition 5.5.

Proof.

6. The Contraction Hypothesis

Definition 6.1 (The Contraction Hypothesis).

Remark 6.2.

Theorem 6.3.

Proof.

7. Final Proof

Proof of Theorem 1.4.

8. Concluding Remarks

References