For $ \beta>1 $, let $ T_\beta $ be the $\beta $-transformation on $ [0,1) $. Let $ \beta _1,\ldots ,\beta _d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. Define

$$ \begin{align*}W(\mathcal P)=\{\mathbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_d})^n(\mathbf{x})\in P_n\text{ infinitely often}\}.\end{align*} $$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}}$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}} W(\mathcal P)$

For given $\epsilon>0$ and $b\in \mathbb {R}^m$, we say that a real $m\times n$ matrix A is $\epsilon $-badly approximable for the target b if

$$ \begin{align*}\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b\rangle^m \geq \epsilon,\end{align*} $$

$\langle \cdot \rangle $

$\epsilon $

$\epsilon $

$\epsilon $

For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9) 106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys. 43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be the continued fraction expansion of an irrational number $x\in [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all $x\in [0,1)$,

$$ \begin{align*} \liminf_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n} = 0\quad \text{and}\quad \limsup_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n}=1. \end{align*} $$

We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.

We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2) 154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets by differentiable real maps. Math. Z. 303 (2023), 3], to prove that under the right hypothesis for the Cantor sets $K_1,\ldots ,K_n$ and the function $h:\mathbb {C}^{n}\to \mathbb {R}^{l}$, the following formula holds:

$$ \begin{align*}HD(h(K_1\times K_2 \times \cdots\times K_n))=\min \{l,HD(K_1)+\cdots+HD(K_n)\}.\end{align*} $$

For every $n\geq 2$, Bourgain’s constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\mathbb {R}^n$ on which harmonic measure is defined. Jones and Wolff (1988, Acta Mathematica 161, 131–144) proved that $b_2=1$. When $n\geq 3$, Bourgain (1987, Inventiones Mathematicae 87, 477–483) proved that $b_n>0$ and Wolff (1995, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing $b_n<1$. Refining Bourgain’s original outline, we prove that

$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$

$n\geq 3$

$c>0$

$b_3\geq 1\times 10^{-15}$

$b_4\geq 2\times 10^{-26}$

We present an elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.

We consider the attractor $\Lambda $ of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of $\Lambda $ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition the dimension equals to the affinity dimension for a typical choice of the linear-parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

For integers a and $b\geq 2$, let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$. The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero.

This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of “k-automatic” sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a structure, we give an algorithmic description of the box-counting dimension, Hausdorff dimension, and Hausdorff measure of the corresponding subset of the unit box. Applications to definability in model-theoretic expansions of the real additive group are laid out as well.

The purpose of this study is two-fold. First, the Hausdorff dimension formula of the multidimensional multiplicative subshift (MMS) in $\mathbb {N}^d$ is presented. This extends the earlier work of Kenyon et al [Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Th. & Dynam. Sys. 32(5) (2012), 1567–1584] from $\mathbb {N}$ to $\mathbb {N}^d$. In addition, the preceding work of the Minkowski dimension of the MMS in $\mathbb {N}^d$ is applied to show that their Hausdorff dimension is strictly less than the Minkowski dimension. Second, the same technique allows us to investigate the multifractal analysis of multiple ergodic average in $\mathbb {N}^d$. Precisely, we extend the result of Fan et al, [Multifractal analysis of some multiple ergodic averages. Adv. Math. 295 (2016), 271–333] of the multifractal analysis of multiple ergodic average from $\mathbb {N}$ to $\mathbb {N}^d$.

Let $x\in [0,1)$ be an irrational number and let $x=[a_{1}(x),a_{2}(x),\ldots ]$ be its continued fraction expansion with partial quotients $\{a_{n}(x): n\geq 1\}$. Given a natural number m and a vector $(x_{1},\ldots ,x_{m})\in [0,1)^{m},$ we derive the asymptotic behaviour of the shortest distance function

$$ \begin{align*} M_{n,m}(x_{1},\ldots,x_{m})=\max\{k\in \mathbb{N}: a_{i+j}(x_{1})=\cdots= a_{i+j}(x_{m}) \ \text{for}~ j=1,\ldots,k \mbox{ and some } i \mbox{ with } 0\leq i \leq n-k\}, \end{align*} $$

which represents the run-length of the longest block of the same symbol among the first n partial quotients of $(x_{1},\ldots ,x_{m}).$ We also calculate the Hausdorff dimension of the level sets and exceptional sets arising from the shortest distance function.

Strong Turing Determinacy, or ${\mathrm {sTD}}$, is the statement that for every set A of reals, if $\forall x\exists y\geq _T x (y\in A)$, then there is a pointed set $P\subseteq A$. We prove the following consequences of Turing Determinacy (${\mathrm {TD}}$) and ${\mathrm {sTD}}$ over ${\mathrm {ZF}}$—the Zermelo–Fraenkel axiomatic set theory without the Axiom of Choice:

1. (1) ${\mathrm {ZF}}+{\mathrm {TD}}$ implies $\mathrm {wDC}_{\mathbb {R}}$—a weaker version of $\mathrm {DC}_{\mathbb {R}}$.

2. (2) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every set of reals is measurable and has Baire property.

3. (3) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that every uncountable set of reals has a perfect subset.

4. (4) ${\mathrm {ZF}}+{\mathrm {sTD}}$ implies that for every set of reals A and every $\epsilon>0$:

   1. (a) There is a closed set $F\subseteq A$ such that $\mathrm {Dim_H}(F)\geq \mathrm {Dim_H}(A)-\epsilon $, where $\mathrm {Dim_H}$ is the Hausdorff dimension.

   2. (b) There is a closed set $F\subseteq A$ such that $\mathrm {Dim_P}(F)\geq \mathrm {Dim_P}(A)-\epsilon $, where $\mathrm {Dim_P}$ is the packing dimension.

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [22, p. 505, Commun. Math. Phys., 2010] for the Navier–Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal {S}$ of suitable weak solution $u$ belonging to $L^{q}(0,T;L^{p}(\mathbb {R}^{3}))$ for $1\leq \frac {2}{q}+\frac {3}{p}\leq \frac 32$ with $2\leq q<\infty$ and $2< p<\infty$ is at most $\max \{p,q\}(\frac {2}{q}+\frac {3}{p}-1)$ in this system. Secondly, it is shown that $1-2s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $u\in L^{2}(0,T;\dot {H}^{s+1}(\mathbb {R}^{3}))$ for $0\leq s\leq \frac 12$ is zero, whose proof relies on Caffarelli–Silvestre's extension. Inspired by Barker–Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity is under some supercritical regularity.

In this paper, we study the Hausdorff dimension of sets defined by almost convergent binary expansion sequences. More precisely, the Hausdorff dimension of the following set

\begin{align*} \bigg\{x\in[0,1)\;:\;\frac{1}{n}\sum_{k=a}^{a+n-1}x_{k}\longrightarrow\alpha\textrm{ uniformly in }a\in\mathbb{N}\textrm{ as }n\rightarrow\infty\bigg\} \end{align*}

$ \alpha\in[0,1] $

$ \alpha $

Let $ T:[0,1]\to [0,1] $ be an expanding Markov map with a finite partition. Let $ \mu _\phi $ be the invariant Gibbs measure associated with a Hölder continuous potential $ \phi $. For $ x\in [0,1] $ and $ \kappa>0 $, we investigate the size of the uniform approximation set

$$ \begin{align*}\mathcal U^\kappa(x):=\{y\in[0,1]:\text{ for all } N\gg1, \text{ there exists } n\le N, \text{ such that }|T^nx-y|<N^{-\kappa}\}.\end{align*} $$

The critical value of $ \kappa $ such that $ \operatorname {\mathrm {\dim _H}}\mathcal U^\kappa (x)=1 $ for $ \mu _\phi $-almost every (a.e.) $ x $ is proven to be $ 1/\alpha _{\max } $, where $ \alpha _{\max }=-\int \phi \,d\mu _{\max }/\int \log |T'|\,d\mu _{\max } $ and $ \mu _{\max } $ is the Gibbs measure associated with the potential $ -\log |T'| $. Moreover, when $ \kappa>1/\alpha _{\max } $, we show that for $ \mu _\phi $-a.e. $ x $, the Hausdorff dimension of $ \mathcal U^\kappa (x) $ agrees with the multifractal spectrum of $ \mu _\phi $.

We consider the set of elements in a translation of the middle-third Cantor set which can be well approximated by algebraic numbers of bounded degree. A doubling dimensional result is given, which enables one to conclude an upper bound on the dimension of the set in question for a generic translation.