# The high-dimensional Weierstrass functions

Haojie Ren and Weixiao Shen  Department of Mathematics, Technion, Haifa,
Israel  Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan
University, No 2005 Songhu Road, Shanghai, China 200438
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For a real analytic periodic function $\phi:\mathbb{R}\to\mathbb{R}^{d}$, an
integer $b\geq 2$ and $\lambda\in(1/b,1)$, we prove that the box dimension and
the Hausdorff dimension of the graph of the Weierstrass function
$W(x)=\sum_{n=0}^{\infty}{{\lambda}^{n}\phi(b^{n}x)}$ are both equal to

$\min\left\\{\log_{\lambda^{-1}}b,\,1+\left(\,d-q\,\right)\left(1+\log_{b}\lambda\right)\right\\},$

where $q=q(\phi,b,\lambda)$ denotes the maximum dimension of all linear spaces
$V<\mathbb{R}^{d}$ such that the projection $\pi_{V}W$ is Lipschitz.

##  1\. Introduction

We study the graphs of the high-dimensional Weierstrass functions

(1.1)
$W(x)=W^{\phi}_{\lambda,\,b}(x)=\sum\limits_{n=0}^{\infty}{{\lambda}^{n}\phi(b^{n}x)},\quad
x\in\mathbb{R},$

where $b>1$, $1/b<\lambda<1$ and $\phi(x):\mathbb{R}\to\mathbb{R}^{d}$ is a
non-constant $\mathbb{Z}$-periodic Lipschitz function. The famous example by
Weierstrass, $\phi(x)=\cos(2\pi x)$, is a continuous nowhere differentiable
function (see [14]). Barański [2] later extended it naturally to the complex
case with $\phi(x)=e^{2\pi ix}$. Weierstrass and related functions’ graphs are
key subjects in fractal geometry, extensively studied since its inception (see
[6], [7, Chapter 5], [8, Section 8.2], among others).

Set $\text{graph}W=\\{(x,W(x)):x\in[0,1)\\}$. Let $\pi_{V}$ denote the
orthogonal projection to the linear space $V<\mathbb{R}^{d}$. We define
$q=q(\phi,\lambda,b)$ as:

(1.2) $q=\max_{V<\mathbb{R}^{d}}\left\\{\text{dim }V\>:\>\pi_{V}\circ
W^{\phi}_{\lambda,\,b}\text{ is Lipschitz}\right\\}.$

We can now state our main results.

######  Main Theorem 1.

Let $b\geq 2$ be an integer, $1/b<\lambda<1$ and let
$\phi:\mathbb{R}\to\mathbb{R}^{d}$ denote a $\mathbb{Z}$-periodic real-
analytic function. Then

(1.3)
$\text{dim}_{B}\,\text{graph}W=\text{dim}_{H}\,\text{graph}W=\min\left\\{\log_{\lambda^{-1}}b,\,1+\left(d-q\right)\left(1+\log_{b}\lambda\right)\right\\}.$

######  Remark 1.1.

The graph of function $W$ is the attractor of an IFS with contraction rates of
$\frac{1}{b}$ and $\lambda$. Thus, we may expect that the right side of (1.3)
equals the affine dimension
$D_{d}=\min\\{\log_{\lambda^{-1}}b,\,1+d(1+\log_{b}\lambda)\\}$, see §2.1. But
$q>0$ is indeed possible. (For instance, with a $\mathbb{Z}$-periodic analytic
function $W_{0}$, let $\phi(x)=W_{0}(x)-\lambda W_{0}(bx)$, yielding
$W^{\phi}_{\lambda,\,b}(x)=W_{0}(x)$ and $q(\phi,\lambda,b)=d$.) The
smoothness of $\pi_{V}W$ in (1.2) leads to the dimension decrease in (1.3).

Given the integer $b$ and function $\phi$, it is natural to study the values
of $q(\phi,\lambda,b)$ under different parameters $\lambda$ in $(1/b,1)$.
Define,

$p=p(\phi,b)=\min_{\lambda\in(1/b,1)}q(\phi,\lambda,b).$

For this purpose, we present the following result:

######  Main Theorem 2.

Let $b\geq 2$ be an integer, and let $\phi:\mathbb{R}\to\mathbb{R}^{d}$ be a
non-constant $\mathbb{Z}$-periodic $C^{5}$ function. Then $p(\phi,b)<d$ and
there are at most a finite number of sets $\lambda\in(1/b,1)$ such that
$p(\phi,b)<q(\phi,\lambda,b).$

These theorems generalize [33 Main Theorem] to the high dimension case.

Some applications. We studied the complex Weierstrass function $W^{e^{2\pi
ix}}_{\lambda,\,b}$, introduced by Barański [2], as a generalization of the
classical Weierstrass function $W^{\cos(2\pi x)}_{\lambda,\,b}$.

######  Corollary 1.1.

Let $\phi(x)=e^{2\pi ix}$. For any integer $b\geq 2$ and $\lambda\in(1/b,1)$,
we have

$\text{dim}_{B}\,\text{graph}W=\text{dim}_{H}\,\text{graph}W=\min\left\\{\log_{\lambda^{-1}}b,\,3+2\log_{b}\lambda\right\\}.$

We typically expect the function $W$ to be so complex that

$\text{dim}_{B}\,\text{graph}W=\text{dim}_{H}\,\text{graph}W=\min\left\\{\log_{\lambda^{-1}}b,\,1+d+d\log_{b}\lambda\right\\}.$

If $p(\phi,b)=0$, then this will hold for all but finitely many parameters
$\lambda\in(1/b,1)$ by Main Theorem 1, 2. Hence, we may anticipate the
existence of numerous functions $\phi$ with $p(\phi,b)=0$. The following
corollary substantiates our intuition. Set,

(1.4) $X=\left\\{\phi\in
C^{\omega}(\mathbb{R};\,\mathbb{R}^{d})\>:\>\phi\text{ is
}\mathbb{Z}\text{-periodic}\right\\}.$

Given intger $r\geq 1$, let

$\parallel f-g\parallel_{r}=\parallel f-g\parallel_{0}+\sum_{k=1}^{r}\parallel
f^{(k)}-g^{(k)}\parallel_{0}\quad\text{ for }f,g\in X.$

######  Corollary 1.2.

For any integers $b\geq 2$ and $r\geq 1$, the set

(1.5) $K=\left\\{\phi\in X\>:\>p(\phi,b)=0\right\\}$

is $C^{1}$ open and $C^{\omega}$ dense in $X$.

Historical remarks. Let $d=1$. In the pioneering work [6], Besicovitch and
Ursell proved that for functions of the form
$\sum_{n=0}^{\infty}b_{n}^{-\alpha}\phi(b_{n}x)$, the graphs have Hausdorff
dimension is $2-\alpha$ if $b_{n+1}/b_{n}\to\infty$ and $\log b_{n+1}/\log
b_{n}\to 1$ (see [3] for recent developments). A map as described in (1.1) is
readily observed to be Hölder continuous with an exponent of
$\log_{b}1/\lambda$, thereby implying that the box dimension of its graph is
at most $D_{1}$. To obtain lower bounds, investigating anti-Holder properties
is crucial, see [22, 28, 34]. Following this approach, Hu and Lau [19] fully
resolved the box dimension of the Weierstrass function for $d=1$. Przytycki
and Urbański [28] showed that the Hausdorff dimension of the graph of such a
$W$ is strictly greater than one. In [26], it was proved that the Hausdorff
dimension of the graph of $W$ is bounded below by $D_{1}-O(1/\log b)$.

The recent research on Weierstrass functions closely relates to Bernoulli
convolutions $\sum_{n}\pm\gamma^{n}$, another central topic in fractal
geometry. Solomyak [27, 39] established that for almost every
$\gamma\in(1/2,1)$, Bernoulli convolution is absolutely continuous (for
algebraic parameters $\gamma$, see Varjú [41]). Combining with Ledrappier
[24], this implies that the Hausdorff dimension of the graph of Takagi
functions (with $b=2$ and $\phi(x)=\text{dist}(x,\mathbb{Z})$ in (1.1)) equals
$D_{1}$ for almost every $\lambda\in(1/2,1)$. Inspired by [39], Tsujii [40]
studied the absolute continuity of certain specific SRB measures in hyperbolic
systems. Mandelbrot [25] conjectured that the Hausdorff dimension of the graph
of $W$ is equal to $D_{1}$ for $\phi(x)=\cos(2\pi x)$ and all
$\lambda\in(1/b,1)$. Building upon the works of [40, 24], Barański et al. [4]
verified this conjecture for integral $b$ and $\lambda$ close to $1$. Through
careful transversaility estimates, the conjecture was subsequently proven for
integral $b$ and all $\lambda\in(1/b,1)$ in Shen [37]. Additionally, refer to
[44] for the case when $\phi(x)=\sin(2\pi x)$. After the exact dimensional
properties of Bernoulli convolutions were proven in [12] (also see [11]),
Hochman [16] demonstrated that the dimension of Bernoulli convolutions is one
outside a set of packing dimension zero (see Varjú [42] for every
transcendental $\gamma$). Following the insights from [16], the Hausdorff
dimension of the graph of the Takagi function has been fully resolved in [5].
Developing a nonlinear version of the last work, [33] determined the Hausdorff
dimension of the graph of the Weierstrass function for integer $b$ and
analytic function $\phi:\mathbb{R}\to\mathbb{R}$.

When $d>1$, Barański [2] introduced the complex Weierstrass function
$\sum_{n=0}^{\infty}{\lambda}^{n}e^{b^{n}2\pi ix},$ and studied the box
dimension of its graph when $b$ is an integer and $\lambda$ is close to one.
Motivated by this research, Ren [32] explored the box dimension of the graphs
of Weierstrass function for $\mathbb{Z}$-periodic Lipschitz functions
$\phi:\mathbb{R}\to\mathbb{R}^{2}$, considering integer $b>1$ and sufficiently
large $b\lambda^{2}$. See [1] for some discussion for anti-Holder properties.

See [15, 20, 35, 36] for randomized Weierstrass functions, and for the graphs
of interpolation functions, see [21].

Organization. In §2, we introduce our main findings: Theorems A and B.
Subsequently, we utilize them to complete the proof of Main Theorems 1 and 2.
In §3, we will introduce Ledrappier-Young theory and state Theorem A’, a
simplified version of Theorem A. Additionally, we will recall some notation
and properties related to entropy, which will be extensively utilized
throughout our paper. We will prove Theorem B in §6. The rest of the sections
are devoted to proving Theorem A’.

Acknowledgment. The authors are supported by National Key R$\&$D Program of
China, Grant No 2021YFA1003204. Additionally, this work has received support
from the New Cornerstone Science Foundation.

##  2\. Main findings

In this section, we introduce the main findings of this paper and provide
explanations for the ideas presented in these theorems. Subsequently, we
utilize these findings to prove Main Theorems 1 and 2.

###  2.1. Main findings

Throughout this paper, we fix an integer $b\geq 2$. Let
$\varLambda=\\{0,1,\ldots,b-1\\}$,
$\varLambda^{\\#}=\bigcup_{n=1}^{\infty}\varLambda^{n}$ and
$\Sigma=\varLambda^{\mathbb{Z}_{+}}$. Ledrappier [24] applies ergodic theory
techniques in smooth dynamical systems to analyze the graphs of Takagi
functions. This exploration is motivated by the observation: the graph of
function $W^{\phi}_{\lambda,\,b}$ serves as an invariant repeller within the
expanding dynamical system
$T:[0,1)\times\mathbb{R}^{d}\to[0,1)\times\mathbb{R}^{d}$,

(2.1) $T(x,y)=\left(bx\mod 1,\frac{y-\phi(x)}{\lambda}\right).$

The slopes of the strong stable manifolds of this system can be expressed as
follows: for $\textbf{j}=j_{1}j_{2}j_{3}\cdot\cdot\cdot\in\Sigma$, set

(2.2)
$Y(x,\textbf{j})=Y_{\lambda,\,b}^{\phi}(x,\textbf{j})=-\sum\limits_{n=1}^{\infty}{\gamma^{n}\phi^{\prime}\left(\frac{x}{b^{n}}+\frac{j_{1}}{b^{n}}+\frac{j_{2}}{b^{n-1}}+\cdot\cdot\cdot+\frac{j_{n}}{b}\right)},\,\,x\in\mathbb{R}$

where

$\gamma=\frac{1}{b\lambda}\in\left(\frac{1}{b},1\right).$

when $d=1$, the separation properties of the functions $Y(x,\textbf{j})$ are
particularly important in [4, 37, 33]. Specifically, we [33] introduced a soft
separation condition and its degenerated counterpart. To analyze the case
$d\geq 2$, we need to generalize these notations to high-dimensional spaces.

######  Definition 2.1.

Given $\lambda\in(1/b,1)$, we say that a $\mathbb{Z}$-periodic $C^{1}$
function $\phi:\mathbb{R}\to\mathbb{R}^{d}$ satisfies

  * •

the condition (H) if

(2.3)
$Y_{\lambda,\,b}^{\pi_{A}\phi}(x,\,\textbf{j})-Y_{\lambda,\,b}^{\pi_{A}\phi}(x,\,\textbf{i})\nequiv
0,\quad\forall\,\textbf{j}\neq\textbf{i}\in\Sigma$

for any one-dimensional linear space $A<\mathbb{R}^{d}$.

  * •

the condition (H∗) if

$Y_{\lambda,\,b}^{\phi}(x,\,\textbf{j})-Y_{\lambda,\,b}^{\phi}(x,\,\textbf{i})\equiv
0,\quad\forall\,\textbf{j},\,\,\textbf{i}\in\Sigma.$

The condition (H) is regarded as a form of separation property. With this
notation, we extend [33, Theorem B] from $d=1$ to the case $d\geq 1$.
Specifically, we obtain the following.

######  Theorem A.

If an analytic $\mathbb{Z}$-periodic function
$\phi:\mathbb{R}\to\mathbb{R}^{d}$ satisfies the condition (H) for an integer
$b\geq 2$ and $\lambda\in(1/b,1)$, then

$\text{dim}_{B}\,\text{graph}W=\text{dim}_{H}\,\text{graph}W=\min\left\\{\log_{\lambda^{-1}}b,\,1+d+d\log_{b}\lambda\right\\}.$

The ideas of Theorem A. $\text{graph}W$ serves as the invariant repeller for
the expanding map $T$ defined by (2.1). The local inverse of $T$ can be
represented as $g_{i}(x,y)=\left(\frac{x+i}{b},\lambda
y+\phi(\frac{x+i}{b})\right)$ where $i\in\Lambda$. Consequently, the graph of
$W$ also acts as the attractor for the nonlinear IFS
$\\{g_{i}\\}_{i\in\Lambda}$. By Falconer’s findings [9], we anticipate that
the dimension of $\text{graph}W$ equals the self-affine dimension, given by
$\min\left\\{\log_{\lambda^{-1}}b,\,1+d+d\log_{b}\lambda\right\\}$.

Let $\mu$ be the lift of the standard Lebesgue measure on $[0,1)$ to the graph
of $W$. Ledrappier [24] demonstrates, through inverse limits, that for most
$\textbf{j}\in\Sigma$, the measures $\pi_{\textbf{j}}\mu$ are exact
dimensional, with $\text{dim}(\pi_{\textbf{j}}\mu)$ equal to a constant
$\alpha$. Here, $\pi_{\textbf{j}}$ is the projection along the strong unstable
manifold of the dynamical system $T$. Hence, it suffices to show that
$\alpha=\min\\{d,\,\log_{\lambda^{-1}}b\\}$, as stated in Theorem 3.1.

Bárány et al. [5] developed powerful methods to study self-affine measures on
the plane (for recent advances, see [18, 30]), which were used to completely
solve the dimension of Takagi function, i.e.
$\phi(x)=\text{dist}(x,\,\mathbb{Z})$ and $b=2$. Building on their ideas, we
[33] developed methods to handle the nonlinear case and proved Theorem A for
$d=1$. However, this approach encounters challenges when $d>1$. Notably,
Hochman’s Entropy Inverse Theorem [16, Theorem 2.8] plays a crucial role in
[33], but its higher-dimensional version [17, Theorem 2.8] becomes more
intricate. In this paper, we fix a good $\textbf{j}\in\Sigma$. We utilize [17,
Theorem 4.12] to establish an Entropy Inverse Theorem for measures
$\pi_{\textbf{j}}\mu$, as outlined in Theorem 4.1. This theorem yields
saturated properties for the majority of
$\\{\pi_{\textbf{i}}\mu\\}_{\textbf{i}\in\Sigma}$. However, employing [17,
Theorem 2.8] directly only provides local saturated properties for
$\pi_{\textbf{j}}\mu$, which is insufficient for deriving a contradiction in
our setting.

Specifically, assuming the claim of Theorem A’ (see §3.1) fails, we obtain a
sequence of $\textbf{j}\in\Sigma$ such that the corresponding measure
convolution entropy does not increase, as shown in Theorem C (see §5). Thus,
by applying Theorem 4.1, we derive a sequence of linear subspaces
$V<\mathbb{R}^{d}$, such that the majority of measures
$\\{\pi_{\textbf{i}}\mu\\}_{\textbf{i}\in\Sigma}$ satisfy saturated properties
with respect to them. Subsequently, we leverage the compactness of linear
subspaces to find a contradictory direction $V_{0}$, where important insights
from Hochman’s work [17] are indispensable.

######  Remark 2.1.

Given a linear space $V<\mathbb{R}^{d}$, (1.1) and (2.2) imply that

$\pi_{V}\circ
W^{\phi}_{\lambda,\,b}=W^{\pi_{V}\phi}_{\lambda,\,b}\quad\text{and}\quad\pi_{V}\circ
Y^{\phi}_{\lambda,\,b}(x,\,\textbf{j})\equiv
Y^{\pi_{V}\phi}_{\lambda,\,b}(x,\,\textbf{j}).$

Thus, we can regard $\pi_{V}\circ W^{\phi}_{\lambda,\,b}$ as a Weierstrass
function with kernel function $\pi_{V}\phi$, and similarly for $\pi_{V}\circ
Y^{\phi}_{\lambda,\,b}(x,\,\textbf{j})$. In particular, we can apply results
in [33] to $\pi_{V}\circ W^{\phi}_{\lambda,\,b}$ when $\text{dim}V=1$. We will
often use this argument without detailed explanation.

###### Proof of Corollary 1.1.

For $\theta\in\mathbb{R}$, define the projection function
$\hat{\pi}_{\theta}:\mathbb{C}\to\mathbb{R}$ as follows:

$\hat{\pi}_{\theta}(z)=\frac{1}{2}\left(ze^{\theta i}+\overline{z}e^{-\theta
i}\right).$

Considering $\mathbb{C}$ as a real linear space, it suffices to show that
$\hat{\pi}_{\theta}\phi(x)=\cos\left(2\pi x+\theta\right)$ satisfies the
condition (H) by Theorem A. This is supported by [33, Corollary 1.1, Theorem
A]. ∎

The following are some equivalent definitions of $q(\phi,\lambda,b)$ and
$p(\phi,b)$ (see Corollary 2.1), which provide increased convenience for our
analysis. Let

(2.4)
$q^{\prime}(\phi,\lambda,b)=\max_{V<\mathbb{R}^{d}}\left\\{\text{dim}V\>:\>\pi_{V}\phi\text{
satisfies the condition }(\text{H}^{*})\right\\}$

and

(2.5) $p^{\prime}(\phi,b)=\min_{\lambda\in(1/b,1)}q^{\prime}(\phi,\lambda,b).$

The following is another main finding in our paper.

######  Theorem B.

Let $b\geq 2$ be an integer, and let $\phi:\mathbb{R}\to\mathbb{R}^{d}$ be a
non-constant $\mathbb{Z}$-periodic $C^{3}$ function. Then
$p^{\prime}(\phi,b)<d$, and there exist only finitely many $\lambda\in(1/b,1)$
with

$p^{\prime}(\phi,b)<q^{\prime}(\phi,\lambda,b).$

The ideas of Theorem B. We will begin by presenting an equivalent
characterization of the condition (H∗) for $d=1$ using the Fourier
coefficients of $\phi$. This ensures that the degenerate parameter
$\lambda\in(1/b,1)$ is a common zero of a sequence of real analytic series, as
discussed in Proposition 6.1. Therefore, the degenerate linear subspace
$V<\mathbb{R}^{d}$ is the common solution space of a sequence of linear
equations dependent on $\lambda$. Hence, we utilize theories from linear
algebra and complex analysis to complete the proof of Theorem B.

###  2.2. Proof of Main Theorems 1, 2

The following [33, Theorem A] provides the dichotomy between condition (H) and
condition (H∗) for $d=1$.

######  Theorem 2.1.

Assume that $\phi:\mathbb{R}\to\mathbb{R}$ is $\mathbb{Z}$-periodic and of
class $C^{k}$, where $k\in\\{5,6,\ldots,\infty,\omega\\}$. Then exactly one of
the following holds:

  1. (i)

$W_{\lambda,\,b}^{\phi}$ is $C^{k}$ and $\phi$ satisfies the condition (H∗);

  2. (ii)

$W_{\lambda,\,b}^{\phi}$ is not Lipschitz and $\phi$ satisfies the condition
(H).

The following is an application of this theorem.

######  Corollary 2.1.

Let $\phi:\mathbb{R}\to\mathbb{R}^{d}$ be a $\mathbb{Z}$-periodic $C^{k}$
function, where $k\in\\{5,6,\ldots,\infty,\omega\\}$. Then, for any
$\lambda\in(1/b,1)$, we have

$q(\phi,\lambda,b)=q^{\prime}(\phi,\lambda,b).$

###### Proof.

Let $V<\mathbb{R}^{d}$ be a linear space such that $\pi_{V}\phi$ satisfies
condition (H∗) and $\text{dim}V=q^{\prime}$. Given a basis $\\{e_{j}\\}_{1\leq
j\leq q^{\prime}}$ of $V$, each $\pi_{\mathbb{R}e_{j}}\phi$ satisfies the
condition (H∗) due to
$\pi_{\mathbb{R}e_{j}}\circ\pi_{V}=\pi_{\mathbb{R}e_{j}}$ and
$\pi_{\mathbb{R}e_{j}}\circ
Y^{\phi}_{\lambda,\,b}=Y^{\pi_{\mathbb{R}e_{j}}\phi}_{\lambda,\,b}$. This
implies that $\pi_{\mathbb{R}e_{j}}\circ W^{\phi}_{\lambda,\,b}$ is $C^{k}$ by
Theorem 2.1 and $\pi_{\mathbb{R}e_{j}}\circ
W^{\phi}_{\lambda,\,b}=W^{\,\pi_{\mathbb{R}e_{j}}\phi}_{\lambda,\,b}$.
Therefore, $\pi_{V}\circ W^{\phi}_{\lambda,\,b}$ is $C^{k}$, implying
$q^{\prime}\leq q$ by (1.2). Utilizing a similar argument and Theorem 2.1, we
have $q\leq q^{\prime}$. ∎

###### Proof of Main Theorem 2.

Corollary 2.1 implies

$q^{\prime}(\phi,\lambda,b)=q(\phi,\lambda,b)\quad\text{and}\quad
p^{\prime}(\phi,b)=p(\phi,b).$

Combining this with Theorem B, we conclude that Main Theorem 2 holds. ∎

Given a linear space $V<\mathbb{R}^{d}$, denote $V^{\bot}$ by the orthogonal
complement space of $V$.

###### Proof of Main Theorem 1.

In the proof of Corollary 2.1, there is a linear space $V<\mathbb{R}^{d}$ such
that $\text{dim}V=q=q^{\prime}$; additionally, $W_{\lambda,\,b}^{\pi_{V}\phi}$
is analytic, and $\pi_{V}\phi$ satisfies (H∗). Combining this with the
maximality of $q^{\prime}$ and Theorem 2.1, we obtain the following claim.

######  Claim 1.

For any $0\neq a\in V^{\bot}$, (2.3) holds for the linear space
$A=\mathbb{R}a$.

Without loss generality, we may assume that $V=\\{0\\}\times\mathbb{R}^{q}$.
Then there is a function $\psi:\mathbb{R}\to\mathbb{R}^{d-q}$ such that
$\pi_{V^{\bot}}\phi(x)=(\psi(x),0)$. Define the map
$F:[0,1)\times\mathbb{R}^{d}\to[0,1)\times\mathbb{R}^{d}$ by
$F(x,y)=\left(x,y-W_{\lambda,\,b}^{\pi_{V}\phi}(x)\right),$ thus

(2.6)
$F\left(\text{graph}W_{\lambda,\,b}^{\phi}\right)=\left\\{\left(x,W_{\lambda,\,b}^{\psi}(x),0\right)\>:\>x\in[0,1)\right\\}.$

Claim 1 implies that $\psi$ satisfies condition (H). Thus we have

$\text{dim}_{B}\,\text{graph}W_{\lambda,\,b}^{\psi}=\text{dim}_{H}\,\text{graph}W_{\lambda,\,b}^{\psi}=\min\left\\{\log_{\lambda^{-1}}b,\,1+\left(d-q\right)\left(1+\log_{b}\lambda\right)\right\\}$

by Theorem A. This implies that (1.3) holds, since $F$ is a smooth
homeomorphism and (2.6) holds. Finally, the uniqueness of $V$ follows from the
maximality of $q^{\prime}$. ∎

##  3\. Ledrappeier’s theorem and entropy

This section serves as preparation for proving Theorem A.

###  3.1.  Ledrappier-Young Theory

In a metric space $X$, a probability measure $\omega$ is called exact-
dimensional if there exists a constant $\beta\geq 0$, such that for
$\omega$-a.e. $x$,

(3.1) $\lim_{r\to 0}\frac{\log\omega\left(\mathbf{B}(x,r)\right)}{\log
r}=\beta.$

We denote $\text{dim }\omega=\beta$, referring to it as the dimension of
$\omega$. The Ledrappier–Young theory, discussed in [23], examines the exact
dimensional properties of ergodic measures under given smooth diffeomorphisms.
To apply it to the differential homomorphism, one considers the inverse limit
(see e.g. [24, 29, 38]).

In this subsection, we fix $\lambda\in\left(\frac{1}{b},1\right)$, and
consider a $\mathbb{Z}$-periodic $C^{2}$ function
$\phi:\mathbb{R}\to\mathbb{R}^{d}$. We denote by $\mu$ the pushforward measure
of the Lebesgue measure on $[0,1)$ to the graph of $W$, defined as
$x\mapsto(x,W(x))$. It’s worth noting that $\text{supp }\mu=\text{graph}W$ is
invariant under the expanding map $T$ defined by (2.1), and $\mu$ is ergodic
with respect to $T$. Let’s review the Ledrappier–Young results from [24]
regarding $\mu$.

Recall $\Lambda=\\{0,\ldots,b-1\\}$ and $\Sigma=\Lambda^{\mathbb{Z}{+}}$. The
shift map $\sigma:\Sigma\to\Sigma$ is defined as
$(i_{1},i_{2},\ldots)\mapsto(i_{2},i_{3},\dots)$. Let $\nu$ denote the uniform
probability measure on $\Sigma$, and $\nu^{\,\mathbb{Z}+}$ denote the product
(Bernoulli) measure on $\Sigma$. For each $i\in\Lambda$, we define the local
inverse of $T$ as

$g_{i}(x,y)=\left(\frac{x+i}{b},\lambda
y+\phi\left(\frac{x+i}{b}\right)\right).$

Using the inverse limit technique, we define the ”inverse” of $T$ as

$G:[0,1)\times\mathbb{R}^{d}\times\Sigma\to[0,1)\times\mathbb{R}^{d}\times\Sigma\quad
G(x,y,\textbf{i})=\left(g_{i_{1}}(x,y),\sigma(\textbf{i})\right).$

Then $\mu=\frac{1}{b}\sum_{i\in\Lambda}g_{i}\mu.$ By iterating this formula,
we obtain

(3.2) $\mu=\frac{1}{b^{n}}\sum_{\textbf{i}\in\Lambda^{n}}g_{\textbf{i}}\mu,$

where $g_{\textbf{i}}=g_{i_{1}}\circ g_{i_{2}}\circ\ldots g_{i_{n}}$. With
$\textbf{i}\in\Sigma$, detailed calculations show that

$Dg_{i_{1}}\left(\begin{array}[]{l}1\\\
Y(x,\textbf{i})\end{array}\right)=\frac{1}{b}\left(\begin{array}[]{l}1\\\
Y\left(\frac{x+i_{1}}{b},\sigma(\textbf{i})\right)\end{array}\right).$

Hence, $Dg_{i_{n}}g_{i_{n-1}}\ldots g_{i_{1}}$ contracts the vector
$(1,Y(x,\textbf{i}))$ at the exponential rate $-\log b$. Set,

(3.3)
$\Gamma_{\textbf{i}}(x)=\Gamma^{\phi}_{\textbf{i}}(x)=\int_{0}^{x}Y^{\phi}_{\lambda,\,b}(t,\textbf{i})dt.$

For every $y\in\mathbb{R}^{d}$, the function $x\mapsto
y+\Gamma_{\textbf{i}}(x)$ defines the integral curve of the vector field
$(1,Y(t,\textbf{i}))$, passing through $(0,y)$. This establishes a foliation
in $[0,1)\times\mathbb{R}^{d}$ where the leaves are ’parallel’. For any
$\textbf{i}\in\Sigma$, set

(3.4)
$\pi_{\textbf{i}}(x,y)=\pi^{\phi}_{\textbf{i}}(x,y)=y-\Gamma^{\phi}_{\textbf{i}}(x),\quad\quad\text{for
}(x,y)\in[0,1)\times\mathbb{R}^{d}.$

So, $\pi_{\textbf{i}}$ projects $(x,y)$ onto the line $x=0$ along the
foliation $\\{y+\Gamma_{\textbf{i}}(x)\\}_{y\in\mathbb{R}^{d}}$. As [33], we
consider $\pi_{\textbf{i}}$ as the flow projection function for i. Then (3.2)
imlies:

(3.5)
$\pi_{\textbf{i}}\mu=\frac{1}{b^{n}}\sum_{\textbf{a}\in\varLambda^{n}}\pi_{\textbf{i}}g_{\textbf{a}}\mu.$

Given $\textbf{i}=i_{1}i_{2}\cdots i_{n}\in\varLambda^{\\#}$, let
$|\textbf{i}|=n$ be the lengh of i, define the reversed sequence

(3.6) $\textbf{i}^{*}=i_{n}i_{n-1}\cdots i_{1}.$

For $\textbf{j}\in\Sigma$, define $\textbf{i}\textbf{j}:=i_{1}\ldots
i_{n}j_{1}j_{2}\ldots\in\Sigma$. Utilizing the methods of [33, Lemma 3.1], we
obtain the transformation equation:

(3.7)
$\pi_{\textbf{j}}g_{\textbf{i}}(x,y)=\lambda^{|\textbf{i}|}\pi_{{\textbf{i}}^{*}\textbf{j}}(x,y)+\pi_{\textbf{j}}g_{\textbf{i}}(0,0)\quad\quad\forall\,(x,y)\in[0,1)\times\mathbb{R}^{d}.$

Through direct calculation, we establish the following claim:

######  Claim 2.

For any $\textbf{i}\in\Lambda^{n}$ and $\textbf{j},\textbf{w}\in\Sigma$, it
holds that
$\parallel\pi_{\textbf{i}\textbf{j}}-\pi_{\textbf{i}\textbf{w}}\parallel_{\infty}=O(\gamma^{n}).$

The following introduces the Ledrappier-Young formula for $\mu$, which can be
derived by the same methods of [24, Proposition 2]. Consequently, we omit the
detailed proof here.

######  Theorem 3.1.

Let $\phi:\mathbb{R}\to\mathbb{R}^{d}$ be a $\mathbb{Z}$-periodic $C^{2}$
function, then

  1. (1)

$\mu$ is exact dimensional;

  2. (2)

there is a constant $\alpha=\alpha(\phi)\in[0,d]$ such that for
$\nu^{\,\mathbb{Z}_{+}}$-a.e. $\textbf{j}\in\Sigma$, $\pi_{\textbf{j}}\mu$ is
exact dimensional with $\text{dim }\pi_{\textbf{j}}\mu=\alpha$. Furthermore,

(3.8) $\text{dim }\mu=1+(1+\log_{b}\lambda)\alpha.$

The following is a direct corollary of Theorem 3.1.

######  Corollary 3.1.

Let $V<\mathbb{R}^{d}$ be a linear space, and denote
$\alpha_{V}=\alpha(\pi_{V}\phi)$ from Theorem 3.1. Then, for
$\nu^{\,\mathbb{Z}_{+}}$-a.e. $\textbf{j}\in\Sigma$, we have that
$\pi_{V}(\pi_{\textbf{j}}\mu)$ is exact dimensional with $\text{dim
}\pi_{V}(\pi_{\textbf{j}}\mu)=\alpha_{V}$.

###### Proof.

By combining (3.4) with (3.3) and (1.1), we obtain:

$\pi_{V}\circ\pi^{\phi}_{\textbf{j}}\left(x,W^{\phi}(x)\right)=\pi^{\pi_{V}\phi}_{\textbf{j}}\left(x,W^{\pi_{V}\phi}(x)\right)\quad\quad\text{for
}x\in[0,1).$

Consequently,
$\pi_{V}(\pi^{\phi}_{\textbf{j}}\mu^{\phi})=\pi^{\pi_{V}\phi}_{\textbf{j}}(\mu^{\pi_{V}\phi}).$
Thus, our claim holds by Theorem 3.1. ∎

The following lemma provides an upper bound for the dimension of
$\text{graph}W$.

######  Lemma 3.1.

$\overline{\text{dim}}_{B}\,\text{graph}W\leq\min\left\\{\log_{\lambda^{-1}}b,\,1+d+d\log_{b}\lambda\right\\}.$

###### Proof.

From (1.1), it follows that $W$ is a $\log_{b}1/\lambda$-Hölder function. By a
standard argument, as demonstrated in [7, Lemma 5.1.6], we have
$\overline{\text{dim}}_{B}\,\text{graph}W\leq 1+d+d\log_{b}\lambda$. Given
$n\geq 1$, we can cover the graph of $W$ over $[m/b^{n},(m+1)/b^{n})$ by a
ball with length of $O(\lambda^{n})$ for each $0\leq m<b^{n}-1$. Consequently,
$\overline{\text{dim}}_{B}\,\text{graph}W\leq\log_{\lambda^{-1}}b$. ∎

Thus, by applying the mass distribution principle, we simplify Theorem A to:

######  Theorem A’.

Let $b\geq 2$ be an integer and $\lambda\in(1/b,1)$. Let
$\phi:\mathbb{R}\to\mathbb{R}^{d}$ be an analytic $\mathbb{Z}$-periodic
function satisfying condition (H). Then,
$\alpha=\min\\{d,\,\log_{1/\lambda}b\\}$, where $\alpha$ is the constant
defined in Theorem 3.1.

###  3.2. Entropy

In this subsection, we outline fundamental entropy properties that are
frequently used in our paper, often without additional mention. These
properties can be found in [17, Section 3.1].

Let $(\Omega,\mathcal{B})$ be a measurable space. Given a probability measure
$\omega$ on $\Omega$ and a countable partition $\mathcal{Q}\subset\mathcal{B}$
of $\Omega$, we define the entropy as follows:

$H(\omega,\mathcal{Q})=\sum_{Q\in\mathcal{Q}}-\omega(Q)\log_{b}\omega(Q),$

where the common convention $0\log 0=0$ is adopted. We denote by
$\mathcal{Q}(x)$ the member of $\mathcal{Q}$ containing $x$. If
$\omega(\mathcal{Q}(x))>0$, the conditional measure

$\omega_{\mathcal{Q}(x)}(A)=\omega_{x,\,\mathcal{Q}}(A)=\frac{\omega(A\cap\mathcal{Q}(x))}{\omega(\mathcal{Q}(x))}$

is called a $\mathcal{Q}$-component of $\omega$. In this paper we always
assume that $H(\omega,\mathcal{Q})<\infty$. Note that we have the following
upper bound:

(3.9)
$H(\omega,\mathcal{Q})\leq\log_{b}\\#\\{Q\in\mathcal{Q}\>:\>\omega(Q)>0\\}.$

Given another countable partition $\mathcal{D}\subset\mathcal{B}$, we define
the conditional entropy as:

$H(\omega,\mathcal{Q}\mid\mathcal{D})=\sum_{D\in\mathcal{D},\,\omega(D)>0}\omega(D)H(\omega_{D},\mathcal{Q}).$

When $\mathcal{Q}$ is a refinement of $\mathcal{D}$, meaning
$\mathcal{Q}(x)\subset\mathcal{D}(x)$ for each $x\in\Omega$, we have:

$H(\omega,\mathcal{Q}\mid\mathcal{D})=H(\omega,\mathcal{Q})-H(\omega,\mathcal{D}).$

Partitions $\mathcal{D},\,\mathcal{Q}$ are $C$-commensurable if each element
of $\mathcal{D}$ intersects with at most $C$ elements of $\mathcal{Q}$, vise
versa, where $C\geq 1$ is a constant. Notably, when $\mathcal{D}$ and
$\mathcal{Q}$ are $C$-commensurable:

(3.10) $|H(\omega,\mathcal{Q})-H(\omega,\mathcal{D})|=O(C).$

Set,

$\mathcal{Q}\vee\mathcal{D}:=\left\\{Q\cap D\>:\>Q\in\mathcal{Q}\text{ and
}D\in\mathcal{D}\right\\}.$

Let $\omega^{\prime}$ be another Borel probability measure on $\Omega$. For
$t\in[0,1]$, we observe the the Concavity of entropy:

$tH(\omega,\mathcal{Q})+(1-t)H(\omega^{\prime},\mathcal{Q})\leq
H(t\omega+(1-t)\omega^{\prime},\mathcal{Q}),$

and the Concavity of conditional entropy:

$tH(\omega,\mathcal{D}\mid\mathcal{Q})+(1-t)H(\omega^{\prime},\mathcal{D}\mid\mathcal{Q})\leq
H(t\omega+(1-t)\omega^{\prime},\mathcal{D}\mid\mathcal{Q}).$

We collectively refer to the above two well-known inequalities as Concavity,
which will be extensively utilized in our paper.

In contrast, we also have the Convexity of entropy:

(3.11)
$H(t\omega+(1-t)\omega^{\prime},\mathcal{Q})\leq\log_{b}2+tH(\omega,\mathcal{Q})+(1-t)H(\omega^{\prime},\mathcal{Q}).$

In particular, we shall often consider the case where $\Omega=\mathbb{R}^{d}$
and $\mathcal{B}$ denotes the Borel $\sigma$-algebra. Let $\mathcal{L}_{n}$
denote the partition of $\mathbb{R}^{d}$ into $b$-adic intervals of level $n$,
that is:

(3.12)
$\mathcal{L}_{n}=\left\\{\left[\frac{j_{1}}{b^{n}},\,\frac{j_{1}+1}{b^{n}}\right)\times\ldots\times\left[\frac{j_{d}}{b^{n}},\,\frac{j_{d}+1}{b^{n}}\right)\>:\>j_{1},\,j_{2},\ldots
j_{d}\in\mathbb{Z}\right\\}.$

There is a constant $C>0$ with the following properties: for any affine map
$f(x)=ax+c$, where $a>0$, $c\in\mathbb{R}^{d}$ and for any $n\in\mathbb{N}$,
we have

(3.13)
$\left|H(f\omega,\mathcal{L}_{n+[\log_{b}a]})-H(\omega,\mathcal{L}_{n})\right|\leq
C;$

given measurable functions $g,g^{\prime}:\Omega\to\mathbb{R}^{d}$ with
$\sup_{x}|g(x)-g^{\prime}(x)|\leq b^{-n}$, then

(3.14)
$\left|H(g\omega,\mathcal{L}_{n})-H(g^{\prime}\omega,\mathcal{L}_{n})\right|\leq
C.$

We denote by $\mathscr{P}(\mathbb{R}^{d})$ the set of all compactly supported
Borel probability measures on $\mathbb{R}^{d}$. For an exact dimensional
probability measure $\omega\in\mathscr{P}(\mathbb{R}^{d})$, its dimension is
closely related to the entropy, as demonstrated in the following fact (cf.
[43, Theorem 4.4]). See also [10, Theorem 1.3].

######  Proposition 3.1.

If $\omega\in\mathscr{P}(\mathbb{R}^{d})$ is exact dimensional, then

$\text{dim
}\omega=\lim\limits_{n\to\infty}\frac{1}{n}H(\omega,\mathcal{L}_{n}).$

Utilizing Proposition 3.1, Theorem 3.1, and the Lebesgue Dominated Convergence
Theorem yields:

(3.15)
$\lim_{n\to\infty}\int\frac{1}{n}H(\pi_{\textbf{j}}\mu,\mathcal{L}_{n})d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})=\alpha.$

###  3.3. Component measures

Let’s consider a sequence of partitions $\mathcal{Q}_{i}$, $i=1,2,\cdots$,
where each $\mathcal{Q}_{i+1}$ refines $\mathcal{Q}_{i}$. We denote
$\omega_{x,\,i}:=\omega_{\mathcal{Q}_{i}(x)}$ as $i$-th component measure of
$\omega$. For a finite set $I$ of positive integers, suppose for each $i\in
I$, there exists a random variable $f_{i}$ defined over
$(\Omega,\mathcal{B}(\mathcal{Q}_{i}),\omega)$, where
$\mathcal{B}(\mathcal{Q}_{i})$ is the sub-$\sigma$-algebra of $\mathcal{B}$
generated by $\mathcal{Q}_{i}$. We introduce the following notation:

$\mathbb{P}_{i\in I}(B_{i})=\mathbb{P}_{i\in
I}^{\omega}(B_{i}):=\frac{1}{\\#I}\sum_{i\in I}\omega(B_{i}),$

where $B_{i}$ is an event for $f_{i}$. If $f_{i}$’s are $\mathbb{R}$-valued
random variable, we also use of the notation:

$\mathbb{E}_{i\in I}(f_{i})=\mathbb{E}^{\omega}_{i\in
I}(f_{i}):=\frac{1}{\\#I}\sum_{i\in I}\mathbb{E}(f_{i}).$

For instance:

(3.16)
$H(\omega,\mathcal{Q}_{m+n}\mid\mathcal{Q}_{n})=\mathbb{E}\left(H(\omega_{x,\,n},\mathcal{Q}_{m+n})\right)=\mathbb{E}_{i=n}\left(H(\omega_{x,\,i},\mathcal{Q}_{i+m})\right).$

These notations were extensively utilized in [16, 5].

The following useful formulas are derived from [16, Lemma 3.4]. The proof
follows the same approach as that of [16] and is consequently omitted.

######  Lemma 3.2.

Given $R\geq 1$ and $\omega\in\mathscr{P}(\mathbb{R}^{d})$ with
$\text{diam}\left(\text{supp}(\omega)\right)\leq R$, it holds that for all
$n\geq m\geq 1$,

$\begin{split}\frac{1}{n}H\left(\omega,\mathcal{L}_{n}\right)&=\frac{1}{n}\sum_{i=0}^{n-1}\frac{1}{m}H\left(\omega,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)+O\left(\frac{m+\log
R}{n}\right)\\\ &=\mathbb{E}^{\omega}_{0\leq
i<n}\left(\frac{1}{m}H(\omega_{y,\,i},\mathcal{L}_{i+m})\right)+O\left(\frac{m+\log
R}{n}\right).\end{split}$

##  4\. Entropy inverse theorem

In this section, we will utilize Hochman’s results [17, Theorem 4.12] to
establish an entropy inverse theorem for measures $\pi_{\textbf{j}}\mu$, which
will serve as a cornerstone in proving Theorem A’. Throughout this section, we
fix $\lambda\in(1/b,1)$ and assume that $\phi:\mathbb{R}\to\mathbb{R}^{d}$ is
a $\mathbb{Z}$-periodic $C^{2}$ function.

Hochman [17] introduced the following notations to study the structures of two
measures, where the convolution entropy does not increase.

######  Definition 4.1.

Let $V<\mathbb{R}^{d}$ be a linear space and $\varepsilon>0$. A measure
$\omega\in\mathscr{P}(\mathbb{R}^{d})$ is said to be
$(V,\,\varepsilon)$-concentrated  if there is a translate $V+y$ of $V$ such
that $1-\varepsilon$ of the mass of $\omega$ lies within a
$\varepsilon$-distance of $V+y$.

######  Definition 4.2.

Let $V<\mathbb{R}^{d}$ be a linear space, $\varepsilon>0$ and $m\geq 1$. A
measure $\omega\in\mathscr{P}(\mathbb{R}^{d})$ is
$(V,\varepsilon,m)$-saturated , if

$\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\right)\geq\frac{1}{m}H\left(\pi_{V^{\bot}}\omega,\mathcal{L}_{m}\right)+\text{dim}V-\varepsilon.$

Recall that $\pi_{V}$ denotes the orthographic projection function from
$\mathbb{R}^{d}$ onto the subspace $V$, while $V^{\bot}$ represents the
orthogonal complement of $V$. It’s worth noting that $\alpha=\alpha(\phi)$ is
defined in Theorem 3.1. With these definitions, we can now state the main
result of this section.

######  Theorem 4.1.

For any $R\geq 1,\,\varepsilon>0$ and $m\geq M(R,\varepsilon)$, there exists
$\delta=\delta(R,\varepsilon,m)>0$ such that the following holds for $n\geq
N(R,\varepsilon,m,\delta)$. Consider a measure
$\eta\in\mathscr{P}(\mathbb{R}^{d})$ with
$\text{diam}\left(\text{supp}(\eta)\right)\leq R$. Suppose there exists
$\textbf{j}\in\Sigma$ such that

$\frac{1}{n}H\left(\eta*\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)<\frac{1}{n}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)+\delta$

and

$\left|\frac{1}{n}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)-\alpha\right|\leq\delta.$

Then, there exists a sequence of linear spaces
$V_{0},V_{1},\ldots,V_{n-1}<\mathbb{R}^{d}$ such that

(4.1) $\mathbb{P}^{\eta}_{0\leq i<n}\left(\eta_{x,\,i}\text{ is
}(V_{i},\varepsilon/b^{i})-\text{concentrated}\right)>1-\varepsilon$

and

(4.2)
$\frac{1}{n}\sum_{i=0}^{n-1}\nu^{\,\mathbb{Z}_{+}}\left(\left\\{\textbf{u}\in\Sigma\>:\>\pi_{\textbf{u}}\mu\text{
is }(V_{i},\varepsilon,m)-\text{saturated}\right\\}\right)>1-\varepsilon.$

######  Remark 4.1.

The first assumption is standard in Hochman’s entropy inverse theorem [16, 17,
Theorem 2.8], so we also call Theorem 4.1 entropy inverse theorem for measure
$\pi_{\textbf{j}}\mu$. Additionally, the second condition holds for almost all
$\textbf{j}\in\Sigma$ when $n$ is large enough.

###  4.1. Basic properties

The next lemma is from [17, Lemma 4.1].

######  Lemma 4.1.

For $m\in\mathbb{Z}_{+}$ and $\eta,\omega\in\mathscr{P}(\mathbb{R}^{d})$, we
have

$\frac{1}{m}H\left(\eta*\omega,\mathcal{L}_{m}\right)\geq\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\right)-O(\frac{1}{m}).$

Given a linear space $V<\mathbb{R}^{d}$ and $m\in\mathbb{Z}_{+}$, we define

$\mathcal{L}_{m}^{V}:=\pi^{-1}_{V}\mathcal{L}_{m}$

following [17]. For any $\omega\in\mathscr{P}(\mathbb{R}^{d})$, we have

(4.3)
$H\left(\pi_{V}\omega,\mathcal{L}_{m}\right)=H\left(\omega,\mathcal{L}_{m}^{V}\right).$

It is easy to verify that,

(4.4) $\mathcal{L}_{m}\text{ and
}\mathcal{L}_{m}^{V}\vee\mathcal{L}_{m}^{V^{\bot}}\text{ are
}O(1)-\text{commensurable}.$

This implies:

$\begin{split}H\left(\omega,\mathcal{L}_{m}\right)&=H\left(\omega,\mathcal{L}_{m}^{V}\vee\mathcal{L}_{m}^{V^{\bot}}\right)+O(1)=H\left(\omega,\mathcal{L}_{m}^{V}\vee\mathcal{L}_{m}^{V^{\bot}}\mid\mathcal{L}_{m}^{V}\right)+H\left(\omega,\mathcal{L}_{m}^{V}\right)+O(1)\\\
&=\int
H\left(\omega_{\mathcal{L}_{m}^{V}(x)},\mathcal{L}_{m}^{V}\vee\mathcal{L}_{m}^{V^{\bot}}\right)d\omega(x)+H\left(\omega,\mathcal{L}_{m}^{V}\right)+O(1)\\\
&=\int\left(H\left(\omega_{\mathcal{L}_{m}^{V}(x)},\mathcal{L}_{m}\right)+O(1)\right)d\omega(x)+H\left(\omega,\mathcal{L}_{m}^{V}\right)+O(1).\end{split}$

Thus we obtain:

(4.5)
$H\left(\omega,\mathcal{L}_{m}\right)=H\left(\omega,\mathcal{L}_{m}\mid\mathcal{L}_{m}^{V}\right)+H\left(\omega,\mathcal{L}_{m}^{V}\right)+O(1)\quad\text{for
any linear space }V<\mathbb{R}^{d}.$

This will be frequently used in our paper without further explanation. If
$\omega$ is $(V,\varepsilon,m)$-saturated, meaning
$\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\right)-\frac{1}{m}H\left(\pi_{V^{\bot}}\omega,\mathcal{L}_{m}\right)\geq\text{dim}V-\varepsilon$,
then (4.5) implies:

(4.6)
$\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\mid\mathcal{L}_{m}^{V^{\bot}}\right)\geq\text{dim}V-\varepsilon-O(\frac{1}{m}).$

Combining the concavity of conditional entropy with (4.5), the following can
be proved using similar methods as in [17, Lemma 4.1].

######  Lemma 4.2.

Given a linear space $V<\mathbb{R}^{d}$, for $m\in\mathbb{Z}_{+}$ and
$\eta,\omega\in\mathscr{P}(\mathbb{R}^{d})$, we have

$\frac{1}{m}H\left(\eta*\omega,\mathcal{L}_{m}\mid\mathcal{L}_{m}^{V}\right)\geq\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\mid\mathcal{L}_{m}^{V}\right)-O(\frac{1}{m}).$

###  4.2. Hochman’s results

For $r>0$ and $c,x\in\mathbb{R}^{d}$, define $S_{r}(x):=rx$ and
$T_{c}(x):=x+c$. Given a measure $\omega\in\mathscr{P}(\mathbb{R}^{d})$, we
denote $r\omega:=S_{r}\omega$ for convenience. The following is a direct
consequence of [17, Theorem 4.12].

######  Theorem 4.2 (Hochman).

For any $\varepsilon>0$, let $m\geq M(\varepsilon)$. Then there exists a
$1\leq k\leq K(\varepsilon,m)$ such that for sufficiently large $n\geq
N(\varepsilon,m,k)$, the following holds. For any
$\eta\in\mathscr{P}(\mathbb{R}^{d})$ there is a sequence
$V_{0},\ldots,V_{n-1}$ of subspaces of $\mathbb{R}^{d}$ such that, writing
$\omega=\eta^{*k}$,

$\mathbb{P}^{\omega}_{0\leq i<n}\left(\,S_{b^{i}}(\omega_{y,\,i})\text{ is
}(V_{i},\varepsilon,m)-\text{saturated}\right)>1-\varepsilon$

and

$\mathbb{P}^{\eta}_{0\leq i<n}\left(\,\eta_{y,\,i}\text{ is
}(V_{i},\varepsilon/b^{i})-\text{concentrated}\right)>1-\varepsilon.$

The following lemma is [17, Corollary 4.16].

######  Lemma 4.3.

Let $\omega,\,\eta\in\mathscr{P}(\mathbb{R}^{d})$. Then

$\frac{1}{n}H\left(\omega*(\eta^{*k}),\mathcal{L}_{n}\right)-\frac{1}{n}H\left(\omega,\mathcal{L}_{n}\right)\leq
k\cdot\left(\frac{1}{n}H\left(\omega*\eta,\mathcal{L}_{n}\right)-\frac{1}{n}H\left(\xi,\mathcal{L}_{n}\right)\right)+O(\frac{k}{n}).$

###  4.3. Proof of Theorem 4.1

We will frequently utilize the following fundamental fact in probability
theory in this paper, details are omitted for brevity.

######  Lemma 4.4.

Let $X$ be a non-negative random variable. For any $M\geq 0$ and
$\varepsilon>0$, if $\,\mathbb{E}X\leq M+\varepsilon$ and
$\mathbb{P}\left(X\geq M-\varepsilon\right)\geq 1-\varepsilon$, then

$\mathbb{P}\left(X\leq M+\sqrt{\varepsilon}\right)\geq
1-O\left((1+M)\sqrt{\varepsilon}\right).$

Notation. For $n\in\mathbb{N}$, denote by $\hat{n}$ the unique integer
satisfying

(4.7) $\lambda^{\hat{n}}\leq b^{-n}<\lambda^{\hat{n}-1}.$

Given $m\geq 1$ and $\omega\in\mathscr{P}(\mathbb{R}^{d})$, for all $i\geq 0$
and $y\in\mathbb{R}^{d}$ such that $\omega({\mathcal{L}_{i}(y)})>0$, we write,

(4.8)
$Q^{\omega}_{m}(i,y)=\int\frac{1}{m}H\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{a}}\mu),\mathcal{L}_{m+i}\right)\,d\nu^{\,\mathbb{Z}_{+}}(\textbf{a}).$

Recall
$\omega_{y,\,i}=\frac{1}{\omega({\mathcal{L}_{i}(y)})}\omega|_{\mathcal{L}_{i}(y)}$
and
$\lambda^{\hat{i}}(\pi_{\textbf{a}}\mu)=S_{\lambda^{\hat{i}}}(\pi_{\textbf{a}}\mu)$,
with $S_{\lambda^{\hat{i}}}(x)=\lambda^{\hat{i}}x$. (4.7) implies that

(4.9) $\text{The diameter of
}\text{supp}\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{a}}\mu)\right)\text{
is bounded by }O(1/{b^{i}}).$

Utilizing this and (3.9), we deduce that $Q^{\omega}_{m}(i,y)=O(1)$.

######  Lemma 4.5.

Given $R\geq 1$, consider a measure $\omega\in\mathscr{P}(\mathbb{R}^{d})$
with $\text{diam}\left(\text{supp}(\omega)\right)\leq R$. For any integers
$n>m\geq 1$ and $\textbf{j}\in\Sigma$, it follows that

$\frac{1}{n}H\left(\omega*\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)\geq\mathbb{E}^{\omega}_{0\leq
i<n}\left(Q^{\omega}_{m}(i,y)\right)+O\left(\frac{1}{m}+\frac{mR}{n}\right).$

###### Proof.

For any $0\leq i<n$, let’s recall that
$\omega=\mathbb{E}_{i}^{\omega}(\omega_{y,\,i})$ and
$\pi_{\textbf{j}}\mu=\frac{1}{b^{\hat{i}}}\sum_{\textbf{z}\in\varLambda^{\hat{i}}}\pi_{\textbf{j}}g_{\textbf{z}}\mu$.
Combining this with Lemma 3.2 and the concavity, we can deduce

$\begin{split}\frac{1}{n}&H\left(\omega*\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)=\frac{1}{n}\sum_{i=0}^{n-1}\frac{1}{m}H\left(\omega*\pi_{\textbf{j}}\mu,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)+O\left(\frac{mR}{n}\right)\\\
&\geq\frac{1}{n}\sum_{i=0}^{n-1}\mathbb{E}_{i}^{\omega}\left(\frac{1}{b^{\hat{i}}}\sum_{\textbf{z}\in\varLambda^{\hat{i}}}\frac{1}{m}H\left(\omega_{y,\,i}*\pi_{\textbf{j}}g_{\textbf{z}}\mu,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)\right)+O\left(\frac{mR}{n}\right)\\\
&=\mathbb{E}_{0\leq
i<n}^{\omega}\left(\frac{1}{b^{\hat{i}}}\sum_{\textbf{z}\in\varLambda^{\hat{i}}}\frac{1}{m}H\left(\omega_{y,\,i}*\pi_{\textbf{j}}g_{\textbf{z}}\mu,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)\right)+O\left(\frac{mR}{n}\right).\end{split}$

Since $Q^{\eta}_{m}(i,y)=O(1)$, we only need to prove the following:

######  Claim 3.

For any $y\in\mathbb{R}^{d}$and $m,i>0$ with $\gamma^{\hat{i}}\leq b^{-m}$, it
follows that

$\frac{1}{b^{\hat{i}}}\sum_{\textbf{z}\in\varLambda^{\hat{i}}}\frac{1}{m}H\left(\omega_{y,\,i}*\pi_{\textbf{j}}g_{\textbf{z}}\mu,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)=Q^{\omega}_{m}(i,y)+O(\frac{1}{m}).$

By combining
$\pi_{\textbf{j}}g_{\textbf{z}}(x,y)=\lambda^{\hat{i}}\pi_{\textbf{z}^{*}\textbf{j}}(x,y)+\pi_{\textbf{j}}g_{\textbf{z}}(0,0)$
with (4.9), we obtain:

(4.10)
$\begin{split}\frac{1}{m}&H\left(\omega_{y,\,i}*\pi_{\textbf{j}}g_{\textbf{z}}\mu,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)=\frac{1}{m}H\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{z}^{*}\textbf{j}}\mu),\mathcal{L}_{i+m}\right)+O(\frac{1}{m}).\end{split}$

Since
$\parallel\pi_{\textbf{z}^{*}\textbf{j}}-\pi_{\textbf{z}^{*}\textbf{j}^{\prime}}\parallel_{\infty}=O(\gamma^{\hat{i}})=O(1/b^{m})$
holds for each $\textbf{j}^{\prime}\in\Sigma$, we can conclude that

$\frac{1}{m}H\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{z}^{*}\textbf{j}}\mu),\mathcal{L}_{i+m}\right)=\frac{1}{m}H\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{z}^{*}\textbf{j}^{\prime}}\mu),\mathcal{L}_{i+m}\right)+O(\frac{1}{m})$

by (3.14). By combining this with (4.10), we derive the following
relationship:

$\frac{1}{m}H\left(\omega_{y,\,i}*\pi_{\textbf{j}}g_{\textbf{z}}\mu,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)=\int_{\Sigma}\frac{1}{m}H\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{z}^{*}\textbf{j}^{\prime}}\mu),\mathcal{L}_{m+i}\right)\,d\nu^{\,\mathbb{Z}_{+}}(\textbf{j}^{\prime})+O(\frac{1}{m}).$

Combining this with the definitions of $\nu^{\,\mathbb{Z}_{+}}$, we obtain:

$\begin{split}\frac{1}{b^{\hat{i}}}\sum_{\textbf{z}\in\varLambda^{\hat{i}}}&\frac{1}{m}H\left(\omega_{y,\,i}*\pi_{\textbf{j}}g_{\textbf{z}}\mu,\mathcal{L}_{i+m}\mid\mathcal{L}_{i}\right)=\frac{1}{b^{\hat{i}}}\sum_{\textbf{z}\in\varLambda^{\hat{i}}}\int\frac{1}{m}H\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{z}\textbf{j}^{\prime}}\mu),\mathcal{L}_{m+i}\right)\,d\nu^{\,\mathbb{Z}_{+}}(\textbf{j}^{\prime})+O(\frac{1}{m})\\\
&=\int\frac{1}{m}H\left(\omega_{y,\,i}*\lambda^{\hat{i}}(\pi_{\textbf{a}}\mu),\mathcal{L}_{m+i}\right)\,d\nu^{\,\mathbb{Z}_{+}}(\textbf{a})+O(\frac{1}{m}).\end{split}$

Therefore, Claim 3 holds by the definitions of $Q^{\omega}_{m}(i,y)$. ∎

When convolution entropy fails to increase, it implies the saturated
properties of one measure can be deduced from another. This observation is
utilized to derive (4.2). Specifically, we can express it as follows:

######  Lemma 4.6.

For any $\varepsilon>0$ and $m\geq M(\varepsilon)$, the following holds: given
measures $\omega,\eta\in\mathscr{P}(\mathbb{R}^{d})$, if $\omega$ is
$(V,\varepsilon,m)$-saturated for some linear subspace $V<\mathbb{R}^{d}$, and
if

(4.11)
$\frac{1}{m}H\left(\omega*\eta,\mathcal{L}_{m}\right)\leq\frac{1}{m}H\left(\eta,\mathcal{L}_{m}\right)+\varepsilon,$

then $\eta$ is $(V,3\varepsilon,m)$-saturated.

###### Proof.

Let $m\geq 1$ be such that $O(\frac{1}{m})<\varepsilon/3$. Using (4.5) and
(4.11), we obtain:

(4.12)
$\frac{1}{m}H\left(\eta,\mathcal{L}_{m}\mid\mathcal{L}^{V^{\bot}}_{m}\right)+\frac{1}{m}H\left(\eta,\mathcal{L}^{V^{\bot}}_{m}\right)\geq\frac{1}{m}H\left(\omega*\eta,\mathcal{L}_{m}\right)-\frac{4}{3}\varepsilon.$

Combining the cancavity with (4.6), we get:

$\begin{split}&\frac{1}{m}H\left(\omega*\eta,\mathcal{L}_{m}\right)=\frac{1}{m}H\left(\omega*\eta,\mathcal{L}^{V^{\bot}}_{m}\right)+\frac{1}{m}H\left(\omega*\eta,\mathcal{L}_{m}\mid\mathcal{L}^{V^{\bot}}_{m}\right)+O(\frac{1}{m})\\\
&\geq\frac{1}{m}H\left(\eta,\mathcal{L}^{V^{\bot}}_{m}\right)+\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\mid\mathcal{L}^{V^{\bot}}_{m}\right)-\frac{1}{3}\varepsilon\geq\frac{1}{m}H\left(\eta,\mathcal{L}^{V^{\bot}}_{m}\right)+\text{dim}V-\frac{5}{3}\varepsilon.\end{split}$

By combining this with (4.12) and(4.5), we conclude that Lemma 4.6 holds. ∎

Notations. Following [31, Section 2.1], we introduce the following notations
to enhance the convenience of our proofs. Consider $m\geq 1$ and $k\geq M(m)$,
implying that $k$ is large with respect to $m$. We denote this relationship as
$m\ll k$. The relation $\ll$ is transitive. Thus, $m\ll k\ll n$ implies that
$k\geq M(m)\geq 1$ and $n\geq K(m,\,k)\geq 1$. Similarly, for $\varepsilon>0$
and $0<\delta<\varepsilon(\varepsilon)$ (i.e., $\delta$ is small with respect
to $\varepsilon>0$), we represent this as $\varepsilon\ll\delta^{-1}$.

######  Remark 4.2.

The main proof method in the following consists of repeatedly applying Lemma
4.4. Specifically, in each paragraph, we first verify the conditions of Lemma
4.4, and then draw conclusions. After repeating this process three times, the
proof is completed by Lemma 4.6.

###### Proof of Theorem 4.1.

Given $\varepsilon>0$ and $m\geq 1$ such that $1\leq R\ll\varepsilon^{-1}\ll
m$, let $K=K(\varepsilon^{8},m)\geq 1$ as defined in Theorem 4.2. Set
$\delta=\varepsilon^{8}/K$ and $n\geq 1$ such that $m,K\ll n$. Let
$\textbf{j}\in\Sigma$ and $\eta\in\mathscr{P}(\mathbb{R}^{d})$ satisfy the
assumption of Theorem 4.1. Thus, by Theorem 4.2, there is $1\leq k\leq K$ and
a sequence $V_{0},\ldots,V_{n-1}<\mathbb{R}^{d}$ such that (4.1) holds and

(4.13) $\mathbb{P}^{\omega}_{0\leq i<n}\left(S_{b^{i}}(\omega_{y,\,i})\text{
is }(V_{i},\varepsilon^{8},m)-\text{saturated}\right)>1-\varepsilon^{8},$

where $\omega=\eta^{*k}$. Hence, our task is reduced to proving (4.2).

Combining Lemma 4.3 with the assumptions of Theorem 4.1,
$\delta=\varepsilon^{8}/K$ and $\varepsilon^{-1},K\ll n$, we obtain:

$\begin{split}\frac{1}{n}H\left(\omega*\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)&\leq\frac{1}{n}H\left(\eta*\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)+k\left(\frac{1}{n}H\left(\eta*\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)-\frac{1}{n}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)\right)+O(\frac{k}{n})\\\
&\leq\frac{1}{n}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{n}\right)+\delta+k\delta+\varepsilon^{8}\leq\alpha+O(\varepsilon^{8}).\end{split}$

Combining this with Lemma 4.5 and $R\ll\varepsilon^{-1}\ll m\ll n$, we have

(4.14) $\mathbb{E}^{\omega}_{0\leq
i<n}\left(Q^{\omega}_{m}(i,y)\right)\leq\alpha+O(\varepsilon^{8}).$

By Theorem 3.1 and $1\ll\varepsilon^{-1}\ll m$, we have

(4.15)
$\nu^{\,\mathbb{Z}_{+}}\left(\textbf{z}\in\Sigma\>:\>\left|\frac{1}{m}H\left(\,\pi_{\textbf{z}}\mu,\mathcal{L}_{m}\right)-\alpha\right|\leq\varepsilon^{8}\right)>1-\varepsilon^{8}.$

Combining this with (4.8) and Lemma 4.1, we obtain:

(4.16) $Q^{\omega}_{m}(i,y)\geq\alpha-O(\varepsilon^{8})\quad\quad\text{for
}\omega-\text{ a.e. }y.$

Thus we have
$\mathbb{E}^{\omega}_{i}\left(Q^{\omega}_{m}(i,y)\right)\geq\alpha-O(\varepsilon^{8})$
for each $0\leq i<n$. Combining this with (4.14) and Lemma 4.4, we find a set
$I\subset\\{0,\ldots,n-1\\}$ such that $\frac{\\#I}{n}\geq
1-O(\varepsilon^{4})$ and

(4.17)
$\mathbb{E}^{\omega}_{i}\left(Q^{\omega}_{m}(i,y)\right)\leq\alpha+O(\varepsilon^{4})\quad\quad\text{for
}i\in I.$

Let $i\in I$. Combining (4.16) with (4.17) and Lemma 4.4, we obtain

(4.18)
$\mathbb{P}^{\omega}_{i}\left(Q^{\omega}_{m}(i,y)\leq\alpha+O(\varepsilon^{2})\right)\geq
1-O(\varepsilon^{2})\quad\quad\text{for }i\in I.$

By Markov’s inequality, (4.13) implies that there is a set
$I^{\prime}\subset\\{0,\ldots,n-1\\}$ such that

$\mathbb{P}^{\omega}_{i}\left(\,S_{b^{i}}(\omega_{y,\,i})\text{ is
}(V_{i},\varepsilon^{8},m)-\text{saturated}\right)>1-O(\varepsilon^{4})\quad\quad\text{for
}i\in I^{\prime},$

with $\frac{\\#I^{\prime}}{n}\geq 1-O(\varepsilon^{4})$. Let $J=I\cap
I^{\prime}$. Choose $i\in J$ arbitrarily. Since $1\ll\varepsilon^{-1}$, we can
find a $y_{i}\in\mathbb{R}^{d}$ with
$Q^{\omega}_{m}(i,y_{i})\leq\alpha+O(\varepsilon^{2})$ and
$S_{b^{i}}(\omega_{y_{i},\,i})$ is $(V_{i},\varepsilon^{8},m)$-saturated.
Combining $Q^{\omega}_{m}(i,y_{i})\leq\alpha+O(\varepsilon^{2})$ with (4.15),
Lemma 4.1 and Lemma 4.4, we obtain

$\nu^{\,\mathbb{Z}_{+}}\left(\textbf{z}\in\Sigma\>:\>\frac{1}{m}H\left(S_{b^{i}}(\omega_{\,y_{i},\,i})\ast\pi_{\textbf{z}}\mu,\mathcal{L}_{m}\right)\leq\alpha+O(\varepsilon)\right)>1-O(\varepsilon).$

Combining this with (4.15), we have

$\nu^{\,\mathbb{Z}_{+}}\left(\textbf{z}\in\Sigma\>:\>\frac{1}{m}H\left(S_{b^{i}}(\omega_{\,y_{i},\,i})\ast\pi_{\textbf{z}}\mu,\mathcal{L}_{m}\right)\leq\frac{1}{m}H\left(\,\pi_{\textbf{z}}\mu,\mathcal{L}_{m}\right)+O(\varepsilon)\right)>1-O(\varepsilon).$

Combining this with Lemma 4.6 and the fact that
$S_{b^{i}}(\omega_{\,y_{i},\,i})$ is $(V_{i},\varepsilon^{8},m)$-saturated,
then

$\nu^{\mathbb{Z}_{+}}\left(\textbf{z}\in\Sigma\>:\>\pi_{\textbf{z}}\mu\text{
is
}(V_{i},O(\varepsilon),m)-\text{saturated}\right)>1-O(\varepsilon)\quad\quad\text{for
}i\in J.$

This implies that (4.2) holds, since $\frac{\\#J}{n}\geq
1-(1-\frac{\\#I^{\prime}}{n})-(1-\frac{\\#I^{\prime}}{n})\geq
1-O(\varepsilon^{4})$. ∎

##  5\. Proof of Theorem A’

In this section, let $\lambda\in(1/b,1)$ be fixed, and consider
$\phi:\mathbb{R}\to\mathbb{R}^{d}$ as a $\mathbb{Z}$-periodic analytic
function satisfying the condition (H). Our approach to proving Theorem A’
involves employing the method of contradiction. We start by introducing
Theorem C, which plays a crucial role in deriving a contradiction. Then, we
utilize Hochman’s observation to infer pertinent properties, leading to the
completion of the theorem’s proof.

######  Theorem C.

If $\alpha<\log_{\lambda^{-1}}b$, then there exists $p_{0}=p_{0}(\alpha)>0$
with the following properties: for any $\delta>0$ and $n\geq
N(\alpha,p_{0},\delta)$, there exists a measures
$\eta\in\mathscr{P}(\mathbb{R}^{d})$ and $\textbf{z}\in\Sigma$ such that
$\text{dim}\left(\text{supp}(\eta)\right)\leq R$ and:

  1. (C.1)

$\frac{1}{n}H\left(\eta,\mathcal{L}_{n}\right)\geq p_{0}$,

  2. (C.2)

$\frac{1}{n}H\left(\eta*\pi_{\textbf{z}}\mu,\mathcal{L}_{n}\right)\leq\alpha+\delta$,

  3. (C.3)

$\left|\frac{1}{n}H\left(\pi_{\textbf{z}}\mu,\mathcal{L}_{n}\right)-\alpha\right|<\delta$,

where $R=R(\phi,b,\lambda)>0$ is a constant.

Most of the proofs of Theorem C are not new, and they closely resemble the
details presented in [33]. Therefore, for readers who are not familiar with
[33], we offer a brief proof in the appendix.

###  5.1. Saturated properties of $\pi_{\textbf{j}}\mu$

The subsequent lemma is already included in the proof of [17, Lemma 3.16], but
we provide the details here for completeness.

######  Lemma 5.1.

Let $R\geq 1$, let $V<\mathbb{R}^{d}$ be a linear space. Consider a measure
$\omega\in\mathscr{P}(\mathbb{R}^{d})$ such that
$\text{diam}\left(\text{supp}(\omega)\right)\leq R$. Take integers $m\geq
t\geq 1$. Then, the following holds:

$\mathbb{E}_{0\leq
i<m}^{\omega}\left(\frac{1}{t}H\left(\omega_{y,\,i},\mathcal{L}_{t+i}\mid\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)\geq\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\mid\mathcal{L}_{m}^{V^{\bot}}\right)-O\left(\frac{tR}{m}+\frac{1}{t}\right).$

###### Proof.

Using Lemma 3.2 with
$\mathcal{L}_{i}^{V^{\bot}}=\pi_{V^{\bot}}^{-1}\mathcal{L}_{i}$ and
$(\pi_{V^{\bot}}\omega)_{y,\,i}=(\pi_{V^{\bot}}\omega)_{\mathcal{L}_{i}(y)}$,
we get:

(5.1)
$\frac{1}{m}H\left(\omega,\mathcal{L}^{V^{\bot}}_{m}\right)=\frac{1}{m}H\left(\pi_{V^{\bot}}\omega,\mathcal{L}_{m}\right)=\mathbb{E}_{0\leq
i<m}^{\pi_{V^{\bot}}\omega}\left(\frac{1}{t}H\left((\pi_{V^{\bot}}\omega)_{\mathcal{L}_{i}(y)},\mathcal{L}_{t+i}\right)\right)+O\left(\frac{tR}{m}\right).$

By Lemma 3.2 and (4.5), we have:

$\begin{split}&\frac{1}{m}H\left(\omega,\mathcal{L}_{m}\right)=\mathbb{E}_{0\leq
i<m}^{\omega}\left(\frac{1}{t}H\left(\omega_{\mathcal{L}_{i}(y)},\mathcal{L}_{t+i}\right)\right)+O\left(\frac{tR}{m}\right)\\\
&=\mathbb{E}_{0\leq
i<m}^{\omega}\left(\frac{1}{t}H\left(\omega_{\mathcal{L}_{i}(y)},\mathcal{L}_{t+i}\mid\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)+\mathbb{E}_{0\leq
i<m}^{\omega}\left(\frac{1}{t}H\left(\omega_{\mathcal{L}_{i}(y)},\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)+O\left(\frac{tR}{m}+\frac{1}{t}\right).\end{split}$

Combining this with (5.1), we only need to prove for each $0\leq i<m$:

(5.2)
$\mathbb{E}_{i}^{\pi_{V^{\bot}}\omega}\left(\frac{1}{t}H\left((\pi_{V^{\bot}}\omega)_{\mathcal{L}_{i}(y)},\mathcal{L}_{t+i}\right)\right)\geq\mathbb{E}_{i}^{\omega}\left(\frac{1}{t}H\left(\omega_{\mathcal{L}_{i}(y)},\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)+O(\frac{1}{t}).$

Easy to confirm:
$\left(\pi_{V^{\bot}}\omega\right)_{\mathcal{L}_{i}(y)}=\pi_{V^{\bot}}(\omega_{\pi_{V^{\bot}}^{-1}\mathcal{L}_{i}(y)})=\pi_{V^{\bot}}(\omega_{\mathcal{L}^{V^{\bot}}_{i}(y)})$.
Consequently,

$\begin{split}&\mathbb{E}_{i}^{\pi_{V^{\bot}}\omega}\left(\frac{1}{t}H\left((\pi_{V^{\bot}}\omega)_{\mathcal{L}_{i}(y)},\mathcal{L}_{t+i}\right)\right)=\mathbb{E}_{i}^{\omega}\left(\frac{1}{t}H\left(\omega_{\mathcal{L}^{V^{\bot}}_{i}(y)},\mathcal{L}^{V^{\bot}}_{t+i}\right)\right)\\\
&\geq\mathbb{E}_{i}^{\omega}\left(\frac{1}{t}H\left(\omega_{\mathcal{L}_{i}^{V^{\bot}}\vee\mathcal{L}_{i}^{V}\vee\mathcal{L}_{i}(y)},\mathcal{L}^{V^{\bot}}_{t+i}\right)\right)\end{split}$

by cancavity. Combining this with the convexity of entropy, (5.2) holds. ∎

We deduce the following application from the observation in [17, Theorem
6.11]:

######  Lemma 5.2.

For any linear space $V<\mathbb{R}^{d}$, we have

$\alpha-\alpha_{V^{\bot}}\geq\limsup_{t\to\infty}\limsup_{m\to\infty}\int\mathbb{E}_{0\leq
i<m}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}\mid\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j}).$

We note that $\alpha_{V^{\bot}}=\alpha_{V^{\bot}}(\phi)$ is defined in
Corollary 3.1.

###### Proof.

Using Theorem 3.1 and Lemma 3.2, we obtain:

$\begin{split}\alpha&=\lim_{m\to\infty}\int\frac{1}{m}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{m}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})=\lim_{t\to\infty}\lim_{m\to\infty}\int\left(\frac{1}{m}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{m}\right)+O\left(\frac{t}{m}+\frac{1}{t}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})\\\
&=\lim_{t\to\infty}\lim_{m\to\infty}\int\mathbb{E}_{0\leq
i<m}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j}).\end{split}$

Thus, we aim to prove:

(5.3) $\liminf_{t\to\infty}\liminf_{m\to\infty}\int\mathbb{E}_{0\leq
i<m}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})\geq\alpha_{V^{\bot}}.$

Let $1\ll t\ll m$. By the definition of conditional entropy, we have

$\mathbb{E}^{\pi_{\textbf{j}}\mu}_{i}\left(H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)=H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{t+i}^{V^{\bot}}\mid\mathcal{L}_{i}\right)\quad\quad\text{for
each }i\in\mathbb{N},\,\textbf{j}\in\Sigma.$

Combining this with (3.5) and
$\pi_{\textbf{j}}g_{\textbf{a}}(x,y)=\lambda^{|\textbf{a}|}\pi_{{\textbf{a}}^{*}\textbf{j}}(x,y)+\pi_{\textbf{j}}g_{\textbf{a}}(0,0)$,
we get:

$\begin{split}&\int\mathbb{E}_{0\leq
i<m}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})=\int\frac{1}{m}\sum_{i=0}^{m-1}\frac{1}{t}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{t+i}^{V^{\bot}}\mid\mathcal{L}_{i}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})\\\
&\geq\frac{1}{m}\sum_{i=0}^{m-1}\int\frac{1}{b^{\hat{i}}}\sum_{\textbf{a}\in\Lambda^{\hat{i}}}\frac{1}{t}H\left(\pi_{\textbf{j}}g_{\textbf{a}}\mu,\mathcal{L}_{t+i}^{V^{\bot}}\mid\mathcal{L}_{i}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})\\\
&=\frac{1}{m}\sum_{i=0}^{m-1}\int\frac{1}{b^{\hat{i}}}\sum_{\textbf{a}\in\Lambda^{\hat{i}}}\frac{1}{t}H\left(\pi_{\textbf{a}^{*}\textbf{j}}\mu,\mathcal{L}_{t}^{V^{\bot}}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})+O(\frac{1}{t}).\end{split}$

Combining this with
$\sup_{\textbf{j}^{\prime}\in\Sigma}\parallel\pi_{\textbf{a}^{*}\textbf{j}}-\pi_{\textbf{a}^{*}\textbf{j}^{\prime}}\parallel_{\infty}=O(b^{-t})$
for $\gamma^{\hat{i}}\leq b^{-t}$, we conclude:

$\int\mathbb{E}_{0\leq
i<m}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})\geq\int\frac{1}{t}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}^{V^{\bot}}_{t}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})+O\left(\frac{t}{m}+\frac{1}{t}\right).$

Thus, (5.3) holds, as
$\lim\limits_{t\to\infty}\int\frac{1}{t}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}^{V^{\bot}}_{t}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})=\alpha_{V^{\bot}}$
by Corollary 3.1. ∎

Let $Gr_{\ell}(d)$ denote the Grassmannian of $\ell$-dimensional linear spaces
of $\mathbb{R}^{d}$, where $0\leq\ell\leq d$. The metric is defined as:

$d(V,V^{\prime})=\sup_{a\in\mathbb{R}^{d},\,|a|\leq
1}\left|\pi_{V}(a)-\pi_{V^{\prime}}(a)\right|,\quad\quad\text{for
}V,V^{\prime}\in Gr_{\ell}(d).$

Additionally, note that $Gr_{\ell}(d)$, equipped with this metric, is compact.

######  Lemma 5.3.

Given $V\in Gr_{\ell}(d)$, consider a sequence of integers
$0<m_{1}<m_{2}<\ldots$ and subspaces $V_{1},V_{2},\ldots\in Gr_{\ell}(d)$ such
that $V_{k}\to V$. Then we have:

$\alpha-\alpha_{V^{\bot}}\geq\limsup_{k\to\infty}\int\frac{1}{m_{k}}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{m_{k}}\mid\mathcal{L}_{m_{k}}^{V_{k}^{\bot}}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j}).$

###### Proof.

Given $\varepsilon>0$, let integers $t,k\geq 1$ be such that
$\varepsilon^{-1}\ll t\ll k$. Therefore, by Lemma 5.2, we have:

(5.4) $\alpha-\alpha_{V^{\bot}}+\varepsilon\geq\int\mathbb{E}_{0\leq
i<m_{k}}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}\mid\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j}).$

Now, combining $V^{\bot}_{k}\to V^{\bot}$ with $\varepsilon^{-1}\ll t\ll k$,
(3.14), and Lemma 5.1, we get:

$\begin{split}&\int\mathbb{E}_{0\leq
i<m_{k}}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}\mid\mathcal{L}_{t+i}^{V^{\bot}}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})\\\
&=\int\mathbb{E}_{0\leq
i<m_{k}}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t}\right)-\frac{1}{t}H\left(\pi_{V^{\bot}}\left(b^{i}(\pi_{\textbf{j}}\mu)_{y,\,i}\right),\mathcal{L}_{t}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})+O(\frac{1}{t})\\\
&=\int\mathbb{E}_{0\leq
i<m_{k}}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t}\right)-\frac{1}{t}H\left(\pi_{V_{k}^{\bot}}\left(b^{i}(\pi_{\textbf{j}}\mu)_{y,\,i}\right),\mathcal{L}_{t}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})+O\left(\frac{1}{t}+\frac{t}{m_{k}}\right)\\\
&=\int\mathbb{E}_{0\leq
i<m_{k}}^{\pi_{\textbf{j}}\mu}\left(\frac{1}{t}H\left((\pi_{\textbf{j}}\mu)_{y,\,i},\mathcal{L}_{t+i}\mid\mathcal{L}_{t+i}^{V_{k}^{\bot}}\right)\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})+O\left(\frac{1}{t}+\frac{t}{m_{k}}\right)\\\
&\geq\int\frac{1}{m_{k}}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{m_{k}}\mid\mathcal{L}_{m_{k}}^{V_{k}^{\bot}}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})-\varepsilon.\end{split}$

Combining this with (5.4), then our claim holds. ∎

###  5.2. Proof of Theorem A’

The following lemma is a well-known fact from [16], and detailed proof can be
found in [33, Lemma 5.6].

######  Lemma 5.4.

Let $p_{0}>0$ and $R\geq 1$. For $0<\varepsilon<\varepsilon_{0}(p_{0},R)$ and
$m\geq M(p_{0},\varepsilon,R)$, the following holds: Consider
$\eta\in\mathscr{P}(\mathbb{R}^{d})$ such that
$\text{diam}\left(\text{supp}(\eta)\right)\leq R$. If

$\mathbb{P}^{\eta}_{0\leq i<m}\left(\begin{matrix}\eta_{y,\,i}\text{ is
}(\\{0\\},\varepsilon/b^{i})-\text{concentrated}\end{matrix}\,\right)>1-\varepsilon,$

then $\frac{1}{m}H(\eta,\mathcal{L}_{m})<p_{0}.$

###### Proof of Theorem A’.

We only need to prove the following claim:

######  Claim 4.

Either $\alpha(\phi)=\min\\{d,\log_{\lambda^{-1}}b\\}$, or there exists a
linear space $V<\mathbb{R}^{d}$ such that $0<\text{dim}(V)<d$ and

$\alpha(\phi)-\alpha_{V^{\bot}}(\phi)\geq\text{dim}(V).$

In particular, this claim implies that Theorem A’ holds directly when $d=1$.
For $d>1$, using induction, we assume Theorem A’ holds for dimensions up to
$d-1$. To establish the case for dimension $d$, we assume
$\alpha(\phi)<\min\\{d,\log_{\lambda^{-1}}b\\}$. Let $V<\mathbb{R}^{d}$ be the
linear space as stated in Claim 4.

Considering $\pi_{V^{\bot}}\phi$ as a function with a range in
$\mathbb{R}^{\text{dim}(V^{\bot})}$, satisfying condition (H) because $\phi$
satisfies it, we obtain
$\alpha\left(\pi_{V^{\bot}}\phi\right)=\min\\{\text{dim}(V^{\bot}),\log_{\lambda^{-1}}b\\}$.
Combining this with
$\alpha_{V^{\bot}}(\phi)=\alpha\left(\pi_{V^{\bot}}\phi\right)$ and Claim 4,
we reach a contradiction, confirming Theorem A’ for dimension $d$.

Now, let’s prove Claim 4. Suppose
$\alpha(\phi)<\min\\{d,\log_{\lambda^{-1}}b\\}$. Let $p_{0}>0$ and $R\geq 1$
be as specified in Theorem C. Let $\varepsilon_{k}>0$ be such that
$R\ll\varepsilon_{k}^{-1}$ and $\varepsilon_{k}\to 0$. Then, we define a
sequence of $\delta_{k}>0$ and integers $m_{k},n_{k}\geq 1$, for
$k=1,2,\ldots$, such that $\varepsilon_{k}^{-1}\ll m_{k}\ll\delta_{k}^{-1}\ll
n_{k}$. Consequently, there exist measures
$\eta_{k}\in\mathscr{P}(\mathbb{R}^{d})$ with
$\text{dim}\left(\text{supp}(\eta)\right)\leq R$, and
$\textbf{j}_{k}\in\Sigma$ satisfying (C.1), (C.2) and (C.3) for $n_{k}$ and
$\delta_{k}$ according to Theorem C. Combining this with Theorem 4.1 and
$R\ll\varepsilon^{-1}_{k}\ll m_{k}\ll\delta_{k}^{-1}\ll n_{k}$, we can find a
sequence of linear spaces
$V_{0,\,k},V_{1,\,k},\ldots,V_{n_{k}-1,\,k}<\mathbb{R}^{d}$ such that (4.1)
and (4.2) hold for $\varepsilon_{k}$, $m_{k}$ and $\eta_{k},\,\textbf{j}_{k}$.

When $k\geq 1$, Markov’s inequality, as applied to equations (4.1) and (4.2),
guarantees the existence of a set $I_{k}\subset\\{0,1,\ldots,n_{k}-1\\}$ such
that $\frac{\\#I_{k}}{n_{k}}\geq 1-O(\sqrt{\varepsilon_{k}})$. Furthermore,
for every $i\in I_{k}$, we have

(5.5) $\mathbb{P}^{\eta_{k}}_{i}\left((\eta_{k})_{x,\,i}\text{ is
}(V_{i,\,k},\varepsilon_{k}/b^{i})-\text{concentrated}\right)>1-O(\sqrt{\varepsilon_{k}}),$
(5.6)
$\nu^{\,\mathbb{Z}_{+}}\left(\left\\{\textbf{w}\in\Sigma\>:\>\pi_{\textbf{w}}\mu\text{
is
}(V_{i,\,k},\varepsilon_{k},m_{k})-\text{saturated}\right\\}\right)>1-O(\sqrt{\varepsilon_{k}}).$

If there is a $i\in I_{k}$ such that $V_{i,\,k}=\mathbb{R}^{d}$, then
according to (5.6), we infer $\alpha\geq d-O(\varepsilon_{k})$ since
$\varepsilon_{k}^{-1}\ll m_{k}$ and (3.15). However, this contradicts the
conditions $\alpha<d$ and $1\ll\varepsilon_{k}^{-1}$. Thus, we conclude that
$\text{dim}V_{i,\,k}<d$ for every $i\in I_{k}$. Furthermore, if
$\text{dim}V_{i,\,k}=0$ for any $i\in I_{k}$, then combining (5.5) with
$\frac{\\#I_{k}}{n_{k}}\geq 1-O(\sqrt{\varepsilon_{k}})$ implies that (C.1)
fails for $\eta_{k}$, as per Lemma 5.4. Consequently, $d>1$ and there exists
an $i_{k}\in I_{k}$ such that $0<\text{dim}V_{i_{k},\,k}<d$. Denote
$V_{k}:=V_{i_{k},\,k}$. Subsequently, (5.6) implies

(5.7)
$\int\frac{1}{m_{k}}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{m_{k}}\mid\mathcal{L}_{m_{k}}^{V_{k}^{\bot}}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{j})\geq\text{dim}V_{k}-O(\sqrt{\varepsilon_{k}}).$

Utilizing compactness, we can assume the existence of an integer $0<\ell<d$
and a linear subspace $V\in Gr_{\ell}(d)$ such that $V_{k}\in Gr_{\ell}(d)$
and $V_{k}\to V$. Combining this with (5.7) and Lemma 5.3, we conclude that
Claim 4 holds. ∎

##  6\. Proofs of Theorem B and Corollary 1.2

In this section, we will first provide an equivalent characterization of the
condition (H∗) using Fourier coefficients. Then, we will combine this with the
theories of linear algebra and complex analysis to complete the proof of
Theorem B, and finally conclude with the proof of Corollary 1.2.

###  6.1. Fourier Analysis of (H∗)

The following directly follows from [13, Theorem 2.1].

######  Lemma 6.1.

Given $\lambda\in(1/b,1)$ and a $\mathbb{Z}$-periodic $C^{1}$ function
$\hat{f}:\mathbb{R}\to\mathbb{R}$, the following statements are equivalent:

  1. (i)

$\hat{f}$ satisfies the condition (H∗);

  2. (ii)

There exist $\textbf{i}\neq\textbf{j}\in\Sigma$ such that
$Y^{\hat{f}}_{\lambda,\,b}(x,\,\textbf{i})\equiv
Y^{\hat{f}}_{\lambda,\,b}(x,\,\textbf{j});$

  3. (iii)

there is a real $\mathbb{Z}$-periodic $C^{1}$ function $\psi$ such that
$\hat{f}(x)=\lambda\psi(bx)-\psi(x)+Constant.$

######  Remark 6.1.

(iii) implies
$\psi(x)=\sum_{n=1}^{\infty}\frac{1}{\lambda^{n}}\left(\hat{f}(x/b^{n})-\hat{f}(0)\right)+\psi(0)$
through direct calculation. Consequently, if $\hat{f}$ is $C^{k}$ for some
$k\geq 1$, then $\psi$ is also a $C^{k}$ function.

###### Proof.

Define the function $h_{\textbf{i}}:\mathbb{R}\to\mathbb{R}$ for any
$\textbf{i}\in\Sigma$ as:

(6.1)
$h_{\textbf{i}}(x)=\sum_{n=1}^{\infty}\frac{1}{\lambda^{n}}\left[\,\hat{f}\left(\frac{x+i_{1}+i_{2}b\ldots+i_{n}b^{n-1}}{b^{n}}\right)-\hat{f}\left(\frac{i_{1}+i_{2}b\ldots+i_{n}b^{n-1}}{b^{n}}\right)\,\right].$

Hence, (2.2) implies
$h^{\prime}_{\textbf{i}}(x)\equiv-Y^{\hat{f}}_{\lambda,\,b}(x,\textbf{i})$.
With $h_{\textbf{i}}(0)=0$, we have:

$h_{\textbf{i}}(x)\equiv h_{\textbf{j}}(x)\quad\quad\text{and}\quad\quad
Y^{\hat{f}}_{\lambda,\,b}(x,\textbf{i})\equiv
Y^{\hat{f}}_{\lambda,\,b}(x,\textbf{j})$

equivalent for all $\textbf{i},\textbf{j}\in\Sigma$. Combining this with [13,
Theorem 2.1], our claim holds. ∎

For any $\mathbb{Z}$-periodic continuous function
$\phi:\mathbb{R}\to\mathbb{R}$ and $m\in\mathbb{Z}$, we define the $m$-th
Fourier coefficient of $\phi$ as follows:

(6.2) $a_{m}(\phi)=\int_{0}^{1}\phi(x)e^{-2\pi i\,mx}dx.$

######  Proposition 6.1.

Suppose $\lambda\in(1/b,1)$ and $\phi:\mathbb{R}\to\mathbb{R}$ is a
$\mathbb{Z}$-periodic function with $C^{3}$ continuity. Then $\phi$ satisfies
the condition (H∗) if and only if:

(6.3)
$\sum_{k=0}^{\infty}a_{tb^{k}}(\phi)\cdot\lambda^{-k}=0,\quad\quad\text{for
each }t\in\mathbb{Z}\text{ with }b\nmid t.$

###### Proof.

If $\phi$ satisfies the condition (H∗), then by Lemma 6.1 and Remark 6.1,
there exists a $\mathbb{Z}$-periodic $C^{3}$ function
$\psi:\mathbb{R}\to\mathbb{R}$ such that
$\phi(x)=\lambda\psi(bx)-\psi(x)+Constant$. Comparing Fourier coefficients
after expanding both sides into Fourier series, we get:

$a_{m}(\phi)=\begin{dcases}a_{m/b}(\psi)\cdot\lambda-a_{m}(\psi),&b\mid m;\\\
-a_{m}(\psi),&b\nmid m.\end{dcases}$

Hence, for any integer $t$ such that $b\nmid t$ and $n\in\mathbb{N}$, we have:

$-a_{tb^{n}}(\psi)=a_{tb^{n}}(\phi)+a_{tb^{n-1}}(\phi)\cdot\lambda+\ldots+a_{t}(\phi)\cdot\lambda^{n},$

This implies
$-a_{tb^{n}}(\psi)\cdot\lambda^{-n}=\sum_{k=0}^{n}a_{tb^{k}}(\phi)\cdot\lambda^{-k}$.
By combining this with $|a_{tb^{n}}(\psi)|=O(\frac{1}{tb^{n}})$ and
$\frac{1}{b\lambda}\in(1/b,1)$, (6.3) holds.

If (6.3) holds for every integer $t$ such that $b\nmid t$ and
$n\in\mathbb{N}$, we define

(6.4)
$d_{tb^{n}}=-\left(a_{tb^{n}}(\phi)+a_{tb^{n-1}}(\phi)\cdot\lambda+\ldots+a_{t}(\phi)\cdot\lambda^{n}\right).$

This implies
$d_{tb^{n}}=-\lambda^{n}\sum_{k=n+1}^{\infty}a_{tb^{k}}(\phi)\cdot\lambda^{-k}$.
Given $|a_{tb^{k}}(\phi)|=O(\frac{1}{t^{3}b^{3k}})$, we conclude
$|d_{tb^{k}}|=O(\frac{1}{t^{3}b^{3k}})$. Combining this with (6.4), we obtain
that the $\mathbb{Z}$-periodic function
$\psi(x):=\sum_{n\in\mathbb{Z}\setminus\\{0\\}}d_{n}\cdot e^{2\pi i\,nx}$ is a
real $C^{1}$ function, and $\phi(x)=\lambda\psi(bx)-\psi(x)+Constant$ through
Fourier coefficient analysis. Thus, $\phi(x)$ satisfies the condition (H∗) by
Lemma 6.1. ∎

###  6.2. Proof of Theorem B

In this subsection, we fix a $\mathbb{Z}$-periodic $C^{3}$ function
$\phi:\mathbb{R}\to\mathbb{R}^{d}$. Specifically, we denote
$\phi(x)=(\phi_{1}(x),\phi_{2}(x),\ldots,\phi_{d}(x))$. We list all positive
integers $t$ (such that $b\nmid t$) in ascending order as
$t_{1}<t_{2}<\ldots$.

For any $1\leq j\leq d$ and $m\in\mathbb{Z}+$, we define functions
$R_{j,\,m},\,I_{j,\,m}:[1/b,1]\to\mathbb{R}$ as follows:

(6.5)
$R_{j,\,m}(\lambda)=\sum_{n=0}^{\infty}\text{Re}\left\\{a_{t_{m}b^{n}}(\phi_{j})\right\\}\cdot\lambda^{-n}\quad\quad
I_{j,\,m}(\lambda)=\sum_{n=0}^{\infty}\text{Im}\left\\{a_{t_{m}b^{n}}(\phi_{j})\right\\}\cdot\lambda^{-n},$

where $\text{Re}\\{\cdot\\}$ denotes the real part and $\text{Im}\\{\cdot\\}$
denotes the imaginary part. Because $\phi$ is a $C^{3}$ function, we have
$|a_{t_{m}b^{n}}(\phi_{j})|=O\left(\frac{1}{t_{m}^{3}b^{3n}}\right)$.
Consequently, $R_{j,\,m}$ and $I_{j,\,m}$ are real analytic.

Given $m\in\mathbb{Z}_{+}$ and $\lambda\in[1/b,1]$, let’s define the matrix
$A_{\lambda,\,m}\in M_{d,\,2m}(\mathbb{R})$ as follows:

(6.6)
$A_{\lambda,\,m}=\begin{pmatrix}R_{1,\,1}(\lambda)&\ldots&R_{1,\,m}(\lambda)&I_{1,\,1}(\lambda)&\ldots&I_{1,\,m}(\lambda)\\\
R_{2,\,1}(\lambda)&\ldots&R_{2,\,m}(\lambda)&I_{2,\,1}(\lambda)&\ldots&I_{2,\,m}(\lambda)\\\
\vdots&&\vdots&\vdots&&\vdots\\\
R_{d,\,1}(\lambda)&\ldots&R_{d,\,m}(\lambda)&I_{d,\,1}(\lambda)&\ldots&I_{d,\,m}(\lambda)\end{pmatrix}.$

The kernel space of $A_{\lambda,\,m}$ is defined as:

(6.7) $\text{Ker
}A_{\lambda,\,m}=\left\\{\,y=(y_{1},\ldots,y_{d})\in\mathbb{R}^{d}:\,yA_{\lambda,\,m}=0\,\right\\}.$

We observe that $\text{Ker }A_{\lambda,\,m+1}\subset\text{Ker
}A_{\lambda,\,m}.$ Recall that $q^{\prime}(\phi,\lambda,b)$ is defined by
(2.4).

######  Lemma 6.2.

For any $\lambda\in(1/b,1)$, we have
$q^{\prime}(\phi,\lambda,b)=\text{dim}\left(\cap_{m=1}^{\infty}\text{Ker
}A_{\lambda,\,m}\right).$

###### Proof.

Let $y\in\mathbb{R}^{d}$ such that $\pi_{\mathbb{R}y}\phi$ satisfies the
condition (H∗). Considering $\pi_{\mathbb{R}y}\phi$ as a function with range
in $\mathbb{R}$, we have:

$\sum_{n=0}^{\infty}a_{t_{m}b^{n}}\left(\sum_{j=1}^{d}y_{j}\phi_{j}\right)\cdot\lambda^{-n}=\sum_{j=1}^{d}y_{j}\sum_{n=0}^{\infty}a_{t_{m}b^{n}}(\phi_{j})\cdot\lambda^{-n}=0\quad\quad\forall\,m\in\mathbb{Z}_{+}$

by Proposition 6.1. Combining this with (6.5), we get:

$\sum_{j=1}^{d}y_{j}\cdot
R_{j,\,m}(\lambda)=0\quad\quad\text{and}\quad\quad\sum_{j=1}^{d}y_{j}\cdot
I_{j,\,m}(\lambda)=0,\quad\quad\forall\,m\in\mathbb{Z}_{+}.$

Hence, $y\in\cap_{m=1}^{\infty}\text{Ker }A_{\lambda,\,m}$ by (6.7), implying
$q^{\prime}\leq\text{dim}\left(\cap_{m=1}^{\infty}\text{Ker
}A_{\lambda,\,m}\right)$ by the arbitrariness of $y$ and the definition of
$q^{\prime}$. By Lemma 6.1 and similar methods,
$\pi_{\cap_{m=1}^{\infty}\text{Ker }A_{\lambda,\,m}}(\phi)$ satisfies the
condition (H∗), implying $\text{dim}\left(\cap_{m=1}^{\infty}\text{Ker
}A_{\lambda,\,m}\right)\leq q^{\prime}$ by the maximality of $q^{\prime}$. ∎

Let $m,n\in\mathbb{Z}_{+}$ and $B=(b_{i,\,j})_{n\times m}\in
M_{n,\,m}(\mathbb{R})$ be a matrix. For any $1\leq i_{1}<\ldots<i_{k}\leq n$
and $1\leq j_{1}<\ldots<j_{k^{\prime}}\leq m$, define the submatrix:

(6.8)
$B\left(\\{i_{1},\ldots,i_{k}\\},\\{j_{1},\ldots,j_{k^{\prime}}\\}\right)=\begin{pmatrix}b_{i_{1},\,j_{1}}&\ldots&b_{i_{1},\,j_{k^{\prime}}}\\\
\vdots&\ddots&\vdots\\\
b_{i_{k},\,j_{1}}&\ldots&b_{i_{k},\,j_{k^{\prime}}}\end{pmatrix}.$

For any $1\leq k\leq\min\\{m,n\\}$, let

(6.9)
$L_{k}(B)=\sum_{I\subset\\{1,\ldots,n\\},\,J\subset\\{1,\ldots,m\\},\,\\#I=\\#J=k}\left(\text{det}\,B(I,J)\right)^{2},$

where $\text{det}(\cdot)$ denotes the determinant. The following is a basic
fact in linear algebra.

######  Lemma 6.3.

For any matrix $B\in M_{n,\,m}(\mathbb{R})$, we have:

$n-\text{dim}\left\\{y\in\mathbb{R}^{n}\>:\>yB=0\right\\}=\max_{1\leq
k\leq\min\\{m,n\\}}\left\\{k\>:\>L_{k}(B)\neq 0\right\\}.$

For any $1\leq j\leq d$, define the function
$D_{j}:[1/b,1]\rightarrow\mathbb{R}^{\mathbb{Z}_{+}}$ as follows:

(6.10)
$D_{j}(\lambda)=\left(L_{j}\left(A_{1}(\lambda)\right),\,L_{j}\left(A_{2}(\lambda)\right),\ldots\right),$

where $A_{m}(\lambda)$ is defined by (6.6) (set,
$L_{j}\left(A_{m}(\lambda)\right)=0$ if $j>m$). Recall that
$p^{\prime}(\phi,b)$ is defined by (2.5).

######  Lemma 6.4.

For any $\lambda\in(1/b,1)$, we have

(6.11) $d-q^{\prime}(\phi,\lambda,b)=\max_{1\leq j\leq
d}\left\\{j\>:\>D_{j}(\lambda)\neq 0\right\\}.$

Furthermore, we get:

(6.12) $d-p^{\prime}(\phi,b)=\max_{1\leq j\leq
d,\,\lambda\in(1/b,1)}\left\\{j\>:\>D_{j}(\lambda)\neq 0\right\\}.$

###### Proof.

By (2.5), we only need to prove (6.11). Since $\text{ker
}A_{\lambda,\,m+1}\subset\text{ker }A_{\lambda,\,m}$ holds for any $m\geq 1$,
there exists $m_{0}\geq d$ such that $\text{ker
}A_{\lambda,\,m_{0}}=\cap_{m\geq 1}\text{ker }A_{\lambda,\,m}$. Combining this
with Lemma 6.2, we have

$q^{\prime}(\phi,\lambda,b)=\text{dim }\left(\text{ker
}A_{\lambda,\,m}\right)\quad\text{for any }m\geq m_{0}.$

Combining this with Lemma 6.3, we get

(6.13) $d-q^{\prime}(\phi,\lambda,b)=\max_{1\leq j\leq
d}\left\\{j\>:\>L_{j}\left(A_{\lambda,\,m}\right)\neq
0\right\\},\quad\text{for }m\geq m_{0}.$

This implies that

(6.14) $L_{j}\left(A_{\lambda,\,m}\right)=0\quad\forall m\geq
m_{0},\,j>d-q^{\prime}(\phi,\lambda,b).$

Since $\text{ker }A_{\lambda,\,m_{0}}\subset\text{ker }A_{\lambda,\,m}$ holds
for any $m\leq m_{0}$, we have

$L_{j}\left(A_{\lambda,\,m}\right)=0\quad\forall m\leq
m_{0},\,j>d-q^{\prime}(\phi,\lambda,b)$

by Lemma 6.3. Combining this with (6.14) and (6.13), we conclude that (6.11)
holds. ∎

###### Proof of Theorem B.

Since $\phi$ is non-constant, there exist $1\leq k\leq d$ and
$m_{0}\in\mathbb{Z}_{+}$, $\ell\in\mathbb{N}$ such that
$a_{t_{m_{0}}b^{\ell}}(\phi_{k})\neq 0$. Thus
$\sum_{n=0}^{\infty}a_{t_{m_{0}}b^{n}}(\phi_{k})\cdot\lambda^{-n}\not\equiv
0$. Consequently, there exists $\lambda_{0}\in(1/b,1)$ such that
$D_{1}(\lambda_{0})\neq 0$, implying $p^{\prime}(\phi,b)<d$ by (6.12).

By (6.12) and (6.10), there exists $m_{1}\in\mathbb{Z}_{+}$ and
$\lambda_{1}\in(1/b,1)$ such that
$L_{d-p^{\prime}}(A_{\lambda_{1},\,m_{1}})\neq 0$. Since the function
$F:[1/b,1]\to\mathbb{R}$ defined by
$F(\lambda)=L_{d-p^{\prime}}(A_{\lambda,\,m_{1}})$ is analytic, there are at
most finitely many $\lambda\in[1/b,1]$ such that $D_{d-p^{\prime}}(\lambda)=0$
by (6.10). Combining this with (6.11) and (6.12), the number of elements
$\lambda\in[1/b,1]$ such that $d-q^{\prime}(\phi,\lambda,b)<d-p^{\prime}$ is
at most finite. ∎

###### Proof of Corollary 1.2.

We no longer fix the function $\phi$, and denote notations with a superscript
$\phi$ to indicate their dependence on the choice of $\phi$. By Lemma 6.4, we
have

(6.15) $K=\bigcup_{\lambda\in(1/b,1)}\left\\{\phi\in
X\>:\>D^{\phi}_{d}(\lambda)\neq 0\right\\}.$

To prove that $K$ is open, we need to establish the following claim.

######  Claim 5.

For any $\lambda\in(1/b,1)$, $1\leq j\leq d$, and $m\geq 1$, the linear
functionals
$R^{\phi}_{j,\,m}(\lambda),\,I^{\phi}_{j,\,m}(\lambda):X\to\mathbb{R}$ are
continuous.

Specifically, combining this with (6.9) and (6.10), we conclude that
$\left\\{\phi\in X\,:\,D^{\phi}_{d}(\lambda)\neq 0\right\\}$ is open.
Consequently, $K$ is open as well by (6.15). Now, we prove the claim. Since

$b^{n}\cdot\left|\,a_{t_{m}b^{n}}(\phi_{j})\,\right|\leq\parallel\phi^{\prime}_{j}\parallel_{\infty}\leq\parallel\phi\parallel_{r},\quad\quad\text{for
}n\in\mathbb{N},$

we have

$\left|\sum_{n=0}^{\infty}a_{t_{m}b^{n}}(\phi_{j})\cdot\lambda^{-n}\right|\leq\frac{b\lambda}{b\lambda-1}\parallel\phi\parallel_{r}.$

Combining this with (6.5), then
$R^{\phi}_{j,\,m}(\lambda),\,I^{\phi}_{j,\,m}(\lambda)$ are bounded
$\mathbb{R}$-linear functionals.

Let $\lambda\in(1/b,1)$, and let $\\{e_{j}\\}_{1\leq j\leq d}$ be the standard
basis of $\mathbb{R}^{d}$. Set $[d]=\\{1,\ldots,d\\}$. For any $\phi\in X$, we
define the function $G:\mathbb{R}\to\mathbb{R}$ as

$G(r)=\det\left(A^{\phi(x)-2r\sum_{j=1}^{d}\cos(2\pi
t_{j}x)e_{j}}_{\lambda,\,d}([d],[d])\right).$

Since $G(r)=\det\left(A^{\phi}_{\lambda,\,d}([d],[d])-rI\right)$ is a
polynomial function (with $I$ denoting the identity map), for any
$\varepsilon>0$, there exists $|r|<\varepsilon$ such that $G(r)\neq 0$. Using
(6.15) and the defintion of $D_{d}(\lambda)$, this implies that the function
$\phi(x)-2r\sum_{j=1}^{d}\cos(2\pi t_{j}x)e_{j}\in K$. We conclude that $K$ is
a dense set. ∎

##  7\. Appendix: Proof of Theorem C

In this appendix, we fix $\lambda\in\left(1/b,1\right)$ and a
$\mathbb{Z}$-periodic analytic function $\phi:\mathbb{R}\to\mathbb{R}^{d}$
satisfying the condition (H), where
$\phi(x)=\left(\phi_{1}(x),\phi_{2}(x),\ldots,\phi_{d}(x)\right)$. Without
loss of generality, we set $\phi(0)=0$ for convenience. We assume
$\alpha<\log_{\lambda^{-1}}b$. We extend results from $d=1$ to $d\geq 1$ as
presented in [33]. Since many of these results can be proven with similar
details, we provide statements without proofs. These results are then used to
prove Theorem C.

###  7.1. Transversality

The following statement directly follows from [33, Theorem 5.1].

######  Theorem 7.1.

There exist positive integers $\ell_{0}$, $Q_{0}$, and a constant $\rho_{0}>0$
such that for each $1\leq j\leq d$ and for $\textbf{u},\textbf{v}\in\Sigma$
with $u_{n}\neq v_{n}$, the following holds:

(7.1)
$\sup_{x\in[0,1)}\left|(\Gamma^{\phi_{j}}_{\textbf{u}})^{\prime}(x)-(\Gamma^{\phi_{j}}_{\textbf{v}})^{\prime}(x)\right|\geq\rho_{0}b^{-Q_{0}n},$

and

(7.2) $\sup_{\begin{subarray}{c}I\in\mathcal{L}_{\ell_{0}}\\\
I\subset[0,1)\end{subarray}}\inf_{x\in
I}\left|(\Gamma^{\phi_{j}}_{\textbf{u}})^{\prime}(x)-(\Gamma^{\phi_{j}}_{\textbf{v}})^{\prime}(x)\right|\geq\rho_{0}\sup_{x\in[0,1]}\left|(\Gamma^{\phi_{j}}_{\textbf{u}})^{\prime}(x)-(\Gamma^{\phi_{j}}_{\textbf{v}})^{\prime}(x)\right|.$

Here, $\Gamma^{\phi}_{\textbf{u}}(x)$ is defined by (3.3), sometimes written
as $\Gamma_{\textbf{u}}(x)$ for convenience.

###  7.2. The partitions $\mathcal{L}_{i}^{\mathcal{X}}$

As done in [33], let us define

$\mathcal{X}=\left\\{\pi_{\textbf{j}}\circ
g_{\textbf{i}}\>:\>\textbf{j}\in\Sigma,\,\textbf{i}\in\Lambda^{\\#}\right\\}.$

In this subsection, we establish a nested sequence of partitions
$\mathcal{L}_{i}^{\mathcal{X}}$ of the space $\mathcal{X}$ following the
approach in [33, Section 6]. We then outline several key properties of these
partitions.

Recall that for any $\textbf{j}\in\Sigma$ and $\textbf{i}\in\varLambda^{\\#}$,
we have:

$\pi_{\textbf{j}}g_{\textbf{i}}(x,y)=\lambda^{|\textbf{i}|}(y-\Gamma_{{\textbf{i}}^{*}\textbf{j}}(x))+\pi_{\textbf{j}}g_{\textbf{i}}(0,0).$

Thus, each element of $\mathcal{X}$ can be expressed as
$\lambda^{t}(y-\psi(x))+c$, where $t\in\mathbb{N}$, $c\in\mathbb{R}^{d}$, and
$\psi(x):\mathbb{R}\to\mathbb{R}^{d}$ is an analytic function with
$\psi(0)=0$. We refer to $|\textbf{i}|$ as the height of the map
$\pi_{\textbf{j}}g_{\textbf{i}}$. Define
$\overline{\pi}:\mathcal{X}\to\mathbb{N}\times(\mathbb{R}^{d})^{M+1}$ as
follows:

$\lambda^{t}(y-\psi(x))+c\mapsto\left(t,\,\psi(\frac{1}{M}),\,\psi(\frac{2}{M}),\ldots,\;\psi(1),\;c\right),$

where $M=b^{\ell_{0}}$ and $\ell_{0}$ is obtained from §7.1. Recall
$\mathcal{L}_{n}$ is the $n$-th $b$-adic partition of $\mathbb{R}^{d}$.

######  Definition 7.1.

For each integer $n\geq 1$, $\mathcal{L}_{n}^{\mathcal{X}}$ consists of non-
empty subsets of $\mathcal{X}$ given by:

$\overline{\pi}^{\,-1}\left(\\{t\\}\times I_{1}\times I_{2}\times\ldots\times
I_{M}\times J\right),$

where $t\in\mathbb{N}$, $I_{1},I_{2},\ldots,I_{M}\in\mathcal{L}_{n}$, and
$J\in\mathcal{L}_{n+[t\log_{b}(1/\lambda)]}$.

The partition $\mathcal{L}_{0}^{\mathcal{X}}$ consists of non-empty subsets of
$\mathcal{X}$ represented as:

$\overline{\pi}^{\,-1}\left(\\{t\\}\times\mathbb{R}^{d}\times\ldots\times\mathbb{R}^{d}\times
J\right),$

where $t\in\mathbb{N}$ and $J\in\mathcal{L}_{[t\log_{b}(1/\lambda)]}$.

The following is a generalization of [33, Lemma 6.1], which can be proven
using the same method.

######  Lemma 7.1.

There exists $A>0$ such that any $i\geq 0$, each element of
$\mathcal{L}_{i}^{\mathcal{X}}$ contains at most $A$ elements of
$\mathcal{L}_{i+1}^{\mathcal{X}}$.

Recall that for all $x\in\mathbb{R}$ and $\textbf{i}\in\Sigma$, we have

(7.3)
$\Gamma^{\phi}_{\textbf{i}}(x)=\left(\Gamma^{\phi_{1}}_{\textbf{i}}(x),\ldots,\Gamma^{\phi_{d}}_{\textbf{i}}(x)\right).$

######  Lemma 7.2.

There exists $K>0$ such that if $\pi_{\textbf{j}}g_{\textbf{u}}$ and
$\pi_{\textbf{j}}g_{\textbf{v}}$ belong to the same element of
$\mathcal{L}_{i}^{\mathcal{X}}$, where $\textbf{j}\in\Sigma$,
$\textbf{u},\textbf{v}\in\varLambda^{\hat{\ell}}$, and $i\geq 1$, then for any
$x\in[0,1)$ and $y\in\mathbb{R}^{d}$,

$|\pi_{\textbf{j}}g_{\textbf{u}}(x,y)-\pi_{\textbf{j}}g_{\textbf{v}}(x,y)|\leq
Kb^{-(\ell+i)}.$

###### Proof.

Combining (7.3) with Theorem 7.1, we establish the following for each $1\leq
j\leq d$:

(7.4)
$\sup_{x\in[0,1)}\left|(\Gamma^{\phi_{j}}_{{\textbf{u}}^{*}\textbf{j}})^{\prime}(x)-(\Gamma^{\phi_{j}}_{{\textbf{v}}^{*}\textbf{j}})^{\prime}(x)\right|\leq
2\rho_{0}^{-1}Mb^{-i}.$

by employing techniques akin to [33, (6.1)]. Hence, we can conclude the proof
using methods similar to [33, Lemma 6.2]. ∎

By combining (7.4) with Theorem 7.1, we can prove the following lemma using
methods similar to [33, Lemma 6.3].

######  Lemma 7.3.

There exists a constant $C\in\mathbb{Z}_{+}$ such that for any
$\textbf{u}\neq\textbf{v}\in\varLambda^{\ell}$, $\ell\geq 1$, and
$\textbf{j}\in\Sigma$, we have
$\mathcal{L}_{C\ell}^{\mathcal{X}}(\pi_{\textbf{j}}g_{\textbf{u}})\neq\mathcal{L}_{C\ell}^{\mathcal{X}}(\pi_{\textbf{j}}g_{\textbf{v}})$.

Following [33], let $\xi$ be a discrete probability measure in the space
$\mathcal{X}$ and $\mu$ be a Borel probability measure in $\mathbb{R}^{d}$.
Define $\xi\cdot\mu$ as the Borel probability measure in $\mathbb{R}^{d}$ such
that for any Borel subset $A$ of $\mathbb{R}^{d}$,

$\xi\cdot\mu(A)=\xi\times\mu\left(\left\\{(\Psi,x)\in\mathcal{X}\times\mathbb{R}^{d}\>:\>\Psi(x)\in
A\right\\}\right).$

For each $\mathbf{u}\in\Lambda^{n}$, set
$\mathbf{u}(0)=\frac{u_{1}+\ldots+u_{n}b^{n-1}}{b^{n}}$. By directly
calculating and $\phi(0)=0$, we have

$g_{\mathbf{u}}(0,0)=\left(\,\mathbf{u}(0),W\left(\mathbf{u}(0)\right)\,\right).$

By utilizing this alongside (7.3) and Theorem 7.1, we can follow [33, Lemma
6.4] to prove the subsequent lemma.

######  Lemma 7.4.

For any $r>0$, there exist $p=p(r)>0,\,p_{1}=p_{1}(r)>0$ such that the
following holds if $i$ and $n$ are sufficiently large. Let $\xi$ be a
probability measure supported in an element of $\mathcal{L}_{i}^{\mathcal{X}}$
such that each element in the support of $\xi$ has height $\hat{\ell}$, and
suppose

$\frac{1}{n}H\big{(}\xi,\mathcal{L}_{i+n}^{\mathcal{X}}\big{)}>r.$

Then

$\nu^{\,\hat{i}}\left(\left\\{\textbf{u}\in\Lambda^{\hat{i}}\>:\>\frac{1}{n}H\big{(}\xi\cdot\big{(}\delta_{g_{\textbf{u}}(0,0)}\big{)},\mathcal{L}_{i+n+\ell}\big{)}\geq
p_{1}\right\\}\right)>p.$

###  7.3. Positive local entropy for $\theta_{\ell}^{\,\textbf{j}}$

We need to introduce a discrete measure

$\theta^{\,\textbf{j}}_{\ell}=\frac{1}{b^{\hat{\ell}}}\sum_{\textbf{i}\in\varLambda^{\hat{\ell}}}\delta_{\pi_{\textbf{j}}g_{\textbf{i}}}\in\boldsymbol{\mathscr{P}}(\mathcal{X})$

for each $\ell\in\mathbb{Z}_{+}$ and analyze the entropy of
$\theta^{\,\textbf{j}}_{\ell}$ with respect to the partitions
$\mathcal{L}_{i}^{\mathcal{X}}$ and also the entropy of

$\pi_{\,\textbf{j}}\mu=\theta_{\ell}^{\textbf{j}}.\mu$

with respect to the partitions $\mathcal{L}_{i}$. We start with analyzing the
entropy of $\theta_{\ell}^{\,\textbf{j}}$ with respect to the partitions
$\mathcal{L}_{i}^{\mathcal{X}}$.

######  Lemma 7.5.

For $\nu^{\,\mathbb{Z}_{+}}$-a.e. $\textbf{j}\in\Sigma$,

(7.5)
$\lim_{\ell\to\infty}\frac{1}{\ell}{H}\left(\theta^{\,\textbf{j}}_{\ell},\mathcal{L}_{0}^{\mathcal{X}}\right)=\lim_{\ell\to\infty}\frac{1}{\ell}H(\pi_{\textbf{j}}\mu,\mathcal{L}_{\ell})=\alpha.$

###### Proof.

Define $\pi_{\ell},\pi:\Sigma\;\to\mathbb{R}^{d+1}$, by
$\pi_{\ell}(\textbf{i})=g_{i_{1}i_{2}\cdots i_{\hat{\ell}}}(0,0)$ and
$\pi(\textbf{i})=\lim_{\ell\to\infty}\pi_{\ell}(\textbf{i})$. Direct
calculation shows that $\|\pi_{\ell}-\pi\|_{\infty}=O(b^{-\ell})$. By
utilizing this and employing the same methods as in [33, Lemma 7.1], the claim
holds. ∎

######  Lemma 7.6.

There exists $C\in\mathbb{Z}_{+}$ such that for each $\textbf{j}\in\Sigma$, we
have

$\lim_{\ell\to\infty}\frac{1}{\ell}H\left(\theta^{\,\textbf{j}}_{\ell},\mathcal{L}_{C\ell}^{\mathcal{X}}\right)=\frac{\log
b}{\log\lambda^{-1}}.$

###### Proof.

By Lemma 7.3, there exists $C\in\mathbb{Z}_{+}$ such that for all $\ell\geq 1$
and any two distinct $\textbf{i},\textbf{k}\in\varLambda^{\hat{\ell}}$,
$\pi_{\textbf{j}}g_{\textbf{i}}$ and $\pi_{\textbf{j}}g_{\textbf{k}}$ lie in
distinct elements of $\mathcal{L}_{C\ell}^{\mathcal{X}}$. Therefore,
$H(\theta^{\,\textbf{j}}_{\ell},\mathcal{L}_{C\ell}^{\mathcal{X}})=\hat{\ell}\log
b$. Since $\lim_{\ell\to\infty}\ell/\hat{\ell}=\log_{b}\lambda^{-1}$, the
lemma follows. ∎

For the remainder, let $\mathbf{j}\in\Sigma$ be chosen to satisfy the
conclusions of Lemmas 7.5 and 7.6, and define
$\theta_{\ell}=\theta_{\ell}^{\,\mathbf{j}}$.

###  7.4. Decomposition of entropy

In the following lemma, we decompose the entropy of $\theta_{\ell}$ and
$\pi_{\mathbf{j}}\mu$ into small scales.

######  Lemma 7.7.

For any $\tau>0$, there exists $C^{\prime}(\tau)>0$ such that if positive
integers $n$ and $\ell$ satisfy $\ell>C^{\prime}(\tau)n$, then

(7.6)
$\frac{1}{C\ell}H\left(\theta_{\ell},\mathcal{L}_{C\ell}^{\mathcal{X}}\mid\mathcal{L}_{0}^{\mathcal{X}}\right)\leq\mathbb{E}_{0\leq
i<C\ell}^{\theta_{\ell}}\left[\frac{1}{n}H\left((\theta_{\ell})_{\Psi,\,i},\mathcal{L}_{i+n}^{\mathcal{X}}\right)\right]+\tau,$
(7.7)
$\frac{1}{C\ell}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{(C+1)\ell}\mid\mathcal{L}_{\ell}\right)\geq\mathbb{E}^{\theta_{\ell}}_{0\leq
i<C\ell}\left[\frac{1}{b^{\hat{i}}}\sum_{\textbf{u}\in\Lambda^{\hat{i}}}\frac{1}{n}H\left((\theta_{\ell})_{\Psi,\,i}.g_{\textbf{u}}\mu,\mathcal{L}_{i+n+\ell}\mid\mathcal{L}_{i+\ell}\right)\right]-\tau.$

###### Proof.

Utilizing Lemma 7.1 and following the similar argument as in [16, Lemma 3.4],
we can prove (7.6) and the subsequent results when $\ell/n$ is sufficiently
large:

(7.8)
$\frac{1}{C\ell}H\left(\pi_{\textbf{j}}\mu,\mathcal{L}_{(C+1)\ell}\mid\mathcal{L}_{\ell}\right)\geq\mathbb{E}^{\theta_{\ell}}_{0\leq
i<C\ell}\left[\frac{1}{n}H\left((\theta_{\ell})_{\Psi,\,i}.\mu,\mathcal{L}_{i+n+\ell}\mid\mathcal{L}_{i+\ell}\right)\right]-\tau.$

Note that
$(\theta_{\ell})_{\Psi,\,i}.\mu=\frac{1}{b^{\hat{i}}}\sum_{\textbf{u}\in\Lambda^{\hat{i}}}(\theta_{\ell})_{\Psi,\,i}.g_{\textbf{u}}\mu$.
By the concavity, we have

$H\left((\theta_{\ell})_{\Psi,\,i}.\mu,\mathcal{L}_{i+n+\ell}\mid\mathcal{L}_{i+\ell}\right)\geq\frac{1}{b^{\hat{i}}}\sum_{\textbf{u}\in\Lambda^{\hat{i}}}H\left((\theta_{\ell})_{\Psi,\,i}.g_{\textbf{u}}\mu,\mathcal{L}_{i+n+\ell}\mid\mathcal{L}_{i+\ell}\right).$

Combining this with (7.8), thus (7.7) holds. ∎

Let $\ell\in\mathbb{Z}+$ and $i\in\mathbb{N}$ be given. Consider
$\Psi\in\mathcal{X}$ such that
$\theta_{\ell}(\mathcal{L}_{i}^{\mathcal{X}}(\Psi))>0$. For any
$n\in\mathbb{Z}_{+}$, we define the function $Z_{n}(\Psi,i,\ell)$ as follows:

(7.9)
$Z_{n}(\Psi,i,\ell)=\frac{1}{b^{\hat{i}}}\sum_{\textbf{u}\in\Lambda^{\hat{i}}}\frac{1}{n}H\left(\left((\theta_{\ell})_{\Psi,\,i}\cdot\delta_{g_{\textbf{u}}(0,0)}\right)\ast\left(\Psi
g_{\textbf{u}}\mu\right),\mathcal{L}_{i+n+\ell}\right).$

######  Lemma 7.8.

There is a constant $C^{\prime}>0$ and for each $\tau>0$ there exists
$K(\tau)$ such that when $n\geq K(\tau)$, $i\geq C^{\prime}n$, the following
holds for each $\Psi\in\mathcal{X}$ such that
$\theta_{\ell}(\mathcal{L}_{i}^{\mathcal{X}}(\Psi))>0$:

$\left|\frac{1}{b^{\hat{i}}}\sum_{\textbf{u}\in\Lambda^{\hat{i}}}\frac{1}{n}H((\theta_{\ell})_{\Psi,\,i}.g_{\textbf{u}}\mu,\mathcal{L}_{i+n+\ell}\mid\mathcal{L}_{i+\ell})-Z_{n}(\Psi,i,\ell)\right|<\tau.$

###### Proof.

Given $\tau^{-1}\ll n\ll i$, utilizing Lemma 7.2 and employing similar
techniques as in [33, Lemma 7.7], we establish the following inequality for
each $\textbf{u}\in\Lambda^{\hat{i}}$:

$\left|\frac{1}{n}H(\eta.g_{\textbf{u}}\mu,\mathcal{L}_{i+n+\ell}|\mathcal{L}_{i+\ell})-\frac{1}{n}H((\eta.\delta_{g_{\textbf{u}}(0,0)})\ast(\Psi
g_{\textbf{u}}\mu),\mathcal{L}_{i+n+\ell})\right|<\tau,$

where $\eta=(\theta_{\ell})_{\Psi,\,i}$. By combining this result with (7.9),
our claim is thus established. ∎

######  Lemma 7.9.

For every $\tau>0$, $n\geq K(\tau)\geq 1$, and $i\geq C^{\prime}n$, the
following inequality holds for all $\ell\geq 1$ and $\Psi\in\mathcal{X}$ such
that $\theta_{\ell}(\mathcal{L}_{i}^{\mathcal{X}}(\Psi))>0$:

$Z_{n}(\Psi,i,\ell)\geq\alpha-\tau.$

###### Proof.

Let $n\ll i$ and $\eta=(\theta_{\ell})_{\Psi,\,i}$. Utilizing Lemma 4.1 and
the definition of $Z_{n}$, we obtain:

(7.10)
$Z_{n}(\Psi,i,\ell)\geq\sum_{\Psi^{\prime}\in\text{supp}(\eta)}\eta(\\{\Psi^{\prime}\\})\cdot\frac{1}{b^{\hat{i}}}\sum_{\textbf{u}\in\Lambda^{\hat{i}}}\frac{1}{n}H\left(\Psi^{\prime}g_{\textbf{u}}\mu,\mathcal{L}_{i+n+\ell}\right)-O(\frac{1}{n}).$

Considering $\Psi^{\prime}\in\text{supp}(\eta)$ and (3.7), we find
$\Psi^{\prime}g_{\textbf{u}}(x,y)=\lambda^{\hat{\ell}+\hat{i}}\pi_{\textbf{u}^{*}\textbf{i}}(x,y)+\text{Constant}$
for some $\textbf{i}\in\Sigma$. Combining this with (7.10) and $n\ll i$, we
obtain:

$\begin{split}Z_{n}(\Psi,i,\ell)&\geq\sum_{\Psi^{\prime}\in\text{supp}(\eta)}\eta(\\{\Psi^{\prime}\\})\cdot\frac{1}{b^{\hat{i}}}\sum_{\textbf{u}\in\Lambda^{\hat{i}}}\int\frac{1}{n}H\left(\pi_{\textbf{u}^{*}\textbf{a}}\mu,\mathcal{L}_{n}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{a})-O(\frac{1}{n})\\\
&=\int\frac{1}{n}H\left(\pi_{\textbf{a}}\mu,\mathcal{L}_{n}\right)d\nu^{\,\mathbb{Z}_{+}}(\textbf{a})-O(\frac{1}{n}).\end{split}$

Thus, our claim holds by Theorem 3.1. ∎

###  7.5. Proof of Theorem C

The following presents the weak law of large numbers for the entropy of
measures $\pi_{\textbf{i}}\mu$.

######  Lemma 7.10.

For any $\varepsilon>0$, $n\geq N(\varepsilon),i\geq I(\varepsilon,n)$,

$\inf_{\textbf{a}\in\Sigma}\mathbb{\nu}^{\,i}\left(\left\\{\textbf{u}\in\varLambda^{i}\>:\>\left|\frac{1}{n}H(\pi_{\textbf{u}\textbf{a}}\mu,\mathcal{L}_{n})-\alpha\right|<\varepsilon\right\\}\right)>1-\varepsilon.$

###### Proof.

Given linear space $A<\mathbb{R}^{d}$ with $\text{dim }A=1$, utilizing (2.3),
Theorem 2.1, and [33, Lemma 4.1], we find that the measure
$\pi_{A}(\pi^{\phi}_{\textbf{u}}\mu^{\phi})=\pi^{\,\pi_{A}\phi}_{\textbf{u}}(\mu^{\,\pi_{A}\phi})$
has no atom for each $\textbf{u}\in\Sigma$. Hence, for any interval
$I\subset\mathbb{R}^{d}$, the function
$\textbf{u}\mapsto\pi^{\phi}_{\textbf{u}}\mu^{\phi}(I)$ is continuous.
Consequently, we establish the claim using methods similar to [33, Lemma 4.2].
∎

######  Corollary 7.1.

There exist positive real numbers $r$ and $p_{2}$ such that for all integers
$1\ll n\ll\ell$, the following holds:

$\mathbb{P}_{0\leq
i<C\ell}^{\theta_{\ell}}\left(\frac{1}{n}H\left((\theta_{\ell})_{\Psi,\,i},\mathcal{L}_{i+n}^{\mathcal{X}}\right)>r\right)>p_{2}.$

###### Proof.

Combining (7.6), Lemma 7.5, Lemma 7.6 and $1\ll n\ll\ell$, we have:

(7.11) $\mathbb{E}_{0\leq
i<C\ell}^{\theta_{\ell}}\left(\frac{1}{n}H\left((\theta_{\ell})_{\Psi,\,i},\mathcal{L}_{i+n}^{\mathcal{X}}\right)\right)\geq\frac{1}{2C}(\log_{\lambda^{-1}}b-\alpha).$

Note that by Lemma 7.1,
$\frac{1}{n}H\left((\theta_{\ell})_{\Psi,\,i},\mathcal{L}_{i+n}^{\mathcal{X}}\right)\leq\log
A$. Combining this inequality with (7.11) and using probability theory, our
claim holds for some constants $r,p_{2}>0$. ∎

###### Proof of Theorem C.

Let $K>0$ be determined by Lemma 7.2, and let $r,p_{2}>0$ be from Corollary
7.1. Choose $p=p(r)$ and $p_{1}=p_{1}(r)>0$ according to Lemma 7.4. Define
$R=4K/\lambda^{2}$ and $p_{0}=p_{1}/2$. For any $\delta>0$, select $\tau>0$
such that $\tau\ll\min\\{1,r,p,p_{1},p_{2},\delta\\}$. Choose integers
$I,n,\ell>0$ satisfying $\tau^{-1}\ll n\ll I\ll\ell$.

Given $\tau^{-1}\ll\ell$ and the fact that (7.5) holds for j, therefore

$\frac{1}{C\ell}H(\pi_{\textbf{j}}\mu,\mathcal{L}_{(C+1)\ell}\mid\mathcal{L}_{\ell})\leq\alpha+\tau.$

Combining this with (7.7) and Lemma 7.8, we have

$\mathbb{E}^{\theta_{\ell}}_{0\leq
i<C\ell}\left(Z_{n}\left(\Psi,i,\ell\right)\right)\leq\alpha+O(\tau).$

Combining this with Lemma 7.9 with Lemma 4.4, we have

$\mathbb{P}^{\theta_{\ell}}_{0\leq
i<C\ell}\left(\,Z_{n}(\Psi,i,\ell)\leq\alpha+O(\tau^{1/2})\,\right)\geq
1-O(\tau^{1/2}).$

Combining this with Corollary 7.1, we can fix a
$\Psi\in\text{supp}(\theta_{\ell})$ and $I\leq i\leq n$ such that

$Z_{n}(\Psi,i,\ell)\leq\alpha+O(\tau^{1/2})\quad\quad\text{and}\quad\quad\frac{1}{n}H\left((\theta_{\ell})_{\Psi,\,i},\mathcal{L}_{i+n}^{\mathcal{X}}\right)>r.$

Let $\xi=(\theta_{\ell})_{\Psi,\,i}$ and
$\Psi(x,y)=\lambda^{\hat{\ell}}\pi_{\textbf{i}}(x,y)+Constant$. Employing a
similar argument as in Lemma 7.9, it follows that for each
$\textbf{u}\in\Lambda^{\hat{i}}$,

(7.12) $\frac{1}{n}H\left((\xi.\delta_{g_{\textbf{u}}(0,0)})\ast(\Psi
g_{\textbf{u}}\mu),\mathcal{L}_{i+n+\ell}\right)\geq\frac{1}{n}H(\pi_{\textbf{u}^{*}\textbf{i}}\mu,\mathcal{L}_{n})-O(\frac{1}{n}).$

Combining Lemma 7.10 with the definition of $\nu^{\,\hat{i}}$, we obtain

(7.13)
$\mathbb{\nu}^{\,\hat{i}}\left(\left\\{\textbf{u}\in\varLambda^{\hat{i}}\>:\>\left|\frac{1}{n}H(\pi_{\textbf{u}^{*}\textbf{i}}\mu,\mathcal{L}_{n})-\alpha\right|<\tau\right\\}\right)>1-\tau.$

Combining this with (7.12), we find

$\mathbb{\nu}^{\,\hat{i}}\left(\left\\{\textbf{u}\in\varLambda^{\hat{i}}\>:\>\frac{1}{n}H\left((\xi.\delta_{g_{\textbf{u}}(0,0)})\ast(\Psi
g_{\textbf{u}}\mu),\mathcal{L}_{i+n+\ell}\right)\geq\alpha-O(\tau)\right\\}\right)>1-O(\tau).$

Furthermore, by combining $Z_{n}(\Psi,i,\ell)\leq\alpha+O(\tau^{1/2})$ and
Lemma 4.4, we have

(7.14)
$\mathbb{\nu}^{\,\hat{i}}\left(\left\\{\textbf{u}\in\varLambda^{\hat{i}}\>:\>\frac{1}{n}H\left((\xi.\delta_{g_{\textbf{u}}(0,0)})\ast(\Psi
g_{\textbf{u}}\mu),\mathcal{L}_{i+n+\ell}\right)\leq\alpha+O(\tau^{1/4})\right\\}\right)>1-O(\tau^{1/4}).$

Using $\frac{1}{n}H\left(\xi,\mathcal{L}_{i+n}^{\mathcal{X}}\right)>r$ and
Lemma 7.4, we have

$\mathbb{\nu}^{\,\hat{i}}\left(\left\\{\textbf{u}\in\varLambda^{\hat{i}}\>:\>\frac{1}{n}H\left(\xi.\delta_{g_{\textbf{u}}(0,0)},\mathcal{L}_{i+n+\ell}\right)\geq
p_{1}\right\\}\right)>p.$

Combining this with (7.13) and (7.14), there exists
$\textbf{u}\in\varLambda^{\hat{i}}$ such that

$\frac{1}{n}H\left((\xi.\delta_{g_{\textbf{u}}(0,0)})\ast(\Psi
g_{\textbf{u}}\mu),\mathcal{L}_{i+n+\ell}\right)\leq\alpha+O(\tau^{1/4}),$
$\left|\frac{1}{n}H(\pi_{\textbf{u}^{*}\textbf{i}}\mu,\mathcal{L}_{n})-\alpha\right|<\tau\quad\text{and}\quad\frac{1}{n}H\left(\xi.\delta_{g_{\textbf{u}}(0,0)},\mathcal{L}_{i+n+\ell}\right)\geq
p_{1}.$

This implies that (C.1), (C.2), (C.3) holds for

$\eta:=\lambda^{-\hat{\ell}-\hat{i}}(\xi.\delta_{g_{\textbf{u}}(0,0)})\quad\text{and}\quad\textbf{z}:=\textbf{u}^{*}\textbf{i}.$

Finally Lemma 7.2 implies $\text{diam}\left(\text{supp}(\eta)\right)\leq R$. ∎

## References

  * [1] Roope Anttila, Balázs Bárány, and Antti Käenmäki.  Level sets of prevalent Hölder functions.  arXiv e-prints, page arXiv:2402.08520, February 2024.
  * [2] Krzysztof Barański.  On the complexification of the Weierstrass non-differentiable function.  Ann. Acad. Sci. Fenn. Math., 27(2):325–339, 2002.
  * [3] Krzysztof Barański.  On the dimension of graphs of weierstrass-type functions with rapidly growing frequencies.  Nonlinearity, 25(1):193, 2011.
  * [4] Krzysztof Barański, Balázs Bárány, and Julia Romanowska.  On the dimension of the graph of the classical Weierstrass function.  Adv. Math., 265:32–59, 2014.
  * [5] Balázs Bárány, Michael Hochman, and Ariel Rapaport.  Hausdorff dimension of planar self-affine sets and measures.  Invent. Math., 216(3):601–659, 2019.
  * [6] Abram S Besicovitch and Harold Douglas Ursell.  Sets of fractional dimensions (v): On dimensional numbers of some continuous curves.  In Classics On Fractals, pages 170–179. CRC Press, 2019.
  * [7] Christopher J Bishop and Yuval Peres.  Fractals in probability and analysis, volume 162.  Cambridge University Press, 2017.
  * [8] Kenneth J Falconer.  The geometry of fractal sets cambridge university press. 1985.  1985\.
  * [9] Kenneth J Falconer.  The hausdorff dimension of self-affine fractals.  In Math. Proc. Cambridge Philos. Soc., volume 103, pages 339–350. Cambridge University Press, 1988.
  * [10] Ai-Hua Fan, Ka-Sing Lau, and Hui Rao.  Relationships between different dimensions of a measure.  Monatsh. Math., 135(3):191–201, 2002.
  * [11] De-Jun Feng.  Dimension of invariant measures for affine iterated function systems.  Duke Math. J., 172(4):701–774, 2023.
  * [12] De-Jun Feng and Huyi Hu.  Dimension theory of iterated function systems.  Comm. Pure Appl. Math., 62(11):1435–1500, 2009.
  * [13] Rui Gao and Weixiao Shen.  Low complexity of optimizing measures over an expanding circle map.  arXiv preprint arXiv:2206.05467, 2022.
  * [14] Godefroy Harold Hardy.  Weierstrass’s non-differentiable function.  Trans. Amer. Math. Soc., 17(3):301–325, 1916.
  * [15] Yanick Heurteaux.  Weierstrass functions with random phases.  Trans. Amer. Math. Soc., 355(8):3065–3077, 2003.
  * [16] Michael Hochman.  On self-similar sets with overlaps and inverse theorems for entropy.  Ann. of Math. (2), 180(2):773–822, 2014.
  * [17] Michael Hochman.  On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^{d}$.  arXiv e-prints, page arXiv:1503.09043, March 2015.
  * [18] Michael Hochman and Ariel Rapaport.  Hausdorff dimension of planar self-affine sets and measures with overlaps.  J. Eur. Math. Soc. (JEMS), 24(7):2361–2441, 2022.
  * [19] Tian You Hu and Ka-Sing Lau.  Fractal dimensions and singularities of the Weierstrass type functions.  Trans. Amer. Math. Soc., 335(2):649–665, 1993.
  * [20] Brian R. Hunt.  The Hausdorff dimension of graphs of Weierstrass functions.  Proc. Amer. Math. Soc., 126(3):791–800, 1998.
  * [21] Lai Jiang and Huo-Jun Ruan.  Box dimension of generalized affine fractal interpolation functions.  J. Fractal Geom., 10(3-4):279–302, 2023.
  * [22] James L. Kaplan, John Mallet-Paret, and James A. Yorke.  The Lyapunov dimension of a nowhere differentiable attracting torus.  Ergodic Theory Dynam. Systems, 4(2):261–281, 1984.
  * [23] F. Ledrappier and L.-S. Young.  The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula.  Ann. of Math. (2), 122(3):509–539, 1985.
  * [24] François Ledrappier.  On the dimension of some graphs.  In Symbolic dynamics and its applications (New Haven, CT, 1991), volume 135 of Contemp. Math., pages 285–293. Amer. Math. Soc., Providence, RI, 1992.
  * [25] Benoit B. Mandelbrot.  Fractals: form, chance, and dimension.  W. H. Freeman and Co., San Francisco, CA, revised edition, 1977.  Translated from the French.
  * [26] R. Daniel Mauldin and S. C. Williams.  On the Hausdorff dimension of some graphs.  Trans. Amer. Math. Soc., 298(2):793–803, 1986.
  * [27] Yuval Peres and Boris Solomyak.  Absolute continuity of Bernoulli convolutions, a simple proof.  Math. Res. Lett., 3(2):231–239, 1996.
  * [28] F. Przytycki and M. Urbański.  On the Hausdorff dimension of some fractal sets.  Studia Math., 93(2):155–186, 1989.
  * [29] Min Qian, Jian-Sheng Xie, and Shu Zhu.  Smooth ergodic theory for endomorphisms.  Springer, 2009.
  * [30] Ariel Rapaport.  On self-affine measures associated to strongly irreducible and proximal systems.  arXiv e-prints, page arXiv:2212.07215, December 2022.
  * [31] Ariel Rapaport.  Dimension of diagonal self-affine sets and measures via non-conformal partitions.  arXiv e-prints, page arXiv:2309.03985, September 2023.
  * [32] Haojie Ren.  Box dimension of the graphs of the generalized Weierstrass-type functions.  Discrete Contin. Dyn. Syst., 43(10):3830–3838, 2023.
  * [33] Haojie Ren and Weixiao Shen.  A dichotomy for the Weierstrass-type functions.  Invent. Math., 226(3):1057–1100, 2021.
  * [34] Fraydoun Rezakhanlou.  The packing measure of the graphs and level sets of certain continuous functions.  Math. Proc. Cambridge Philos. Soc., 104(2):347–360, 1988.
  * [35] Julia Romanowska.  Measure and Hausdorff dimension of randomized Weierstrass-type functions.  Nonlinearity, 27(4):787–801, 2014.
  * [36] Alexander Schied and Zhenyuan Zhang.  Weierstrass bridges.  Trans. Amer. Math. Soc., 2024.
  * [37] Weixiao Shen.  Hausdorff dimension of the graphs of the classical Weierstrass functions.  Math. Z., 289(1-2):223–266, 2018.
  * [38] Lin Shu.  Dimension theory for invariant measures of endomorphisms.  Comm. Math. Phys., 298(1):65–99, 2010.
  * [39] Boris Solomyak.  On the random series $\sum\pm\lambda^{n}$ (an Erdös problem).  Ann. of Math. (2), 142(3):611–625, 1995.
  * [40] Masato Tsujii.  Fat solenoidal attractors.  Nonlinearity, 14(5):1011–1027, 2001.
  * [41] Péter P. Varjú.  Absolute continuity of Bernoulli convolutions for algebraic parameters.  J. Amer. Math. Soc., 32(2):351–397, 2019.
  * [42] Péter P. Varjú.  On the dimension of Bernoulli convolutions for all transcendental parameters.  Ann. of Math. (2), 189(3):1001–1011, 2019.
  * [43] Lai Sang Young.  Dimension, entropy and Lyapunov exponents.  Ergodic Theory Dynam. Systems, 2(1):109–124, 1982.
  * [44] Na Zhang.  The hausdorff dimension of the graphs of fractal functions. (in chinese).  Master Thesis., 2018.