Recently, Barros et al. [‘On the shortest distance between orbits and the longest common substring problem’, Adv. Math. 344 (2019), 311–339] adopted a dynamical system perspective to study the decay of the shortest distance between orbits. We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance between orbits in conformal iterated function systems.

We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order r, including negative values of r. To this end, we employ the concept of partition functions, which generalises the notion of the $L^q$-spectrum, thus extending the authors’ earlier work with Sanguo Zhu in a natural way. In particular, we derive inherent fractal-geometric bounds and easily verifiable necessary conditions for the existence of quantization dimensions. We state the exact asymptotics of the quantization error of negative order for absolutely continuous measures, thereby providing an affirmative answer to an open question regarding the geometric mean error posed by Graf and Luschgy in this journal in 2004.

We establish a polynomial ergodic theorem for actions of the affine group of a countable field K. As an application, we deduce—via a variant of Furstenberg’s correspondence principle—that for fields of characteristic zero, any ‘large’ set $E\subset K$ contains ‘many’ patterns of the form $\{p(u)+v,uv\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a finitistic variant of Bergelson’s ‘colouring trick’, show that for $r\in \mathbb N$ fixed, any r-colouring of a large enough finite field will contain monochromatic patterns of the form $\{u,p(u)+v,uv\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ $\{u,v,u+v,uv\}$ theorem.

Let $[a_1(x),a_2(x),a_3(x),\dots ]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. Denote by $S_{n}(x):=\sum _{k=1}^{n} a_{k}(x)$ the sum of partial quotients of x. From the results of Khintchine (1935), Diamond and Vaaler (1986), and Philipp (1988), it follows that for Lebesgue almost every $x \in (0,1)$,

$$\begin{align*}\liminf _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\frac{1}{\log 2} \quad \text {and} \quad \limsup _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\infty. \end{align*}$$

We establish the pointwise equidistribution of self-similar measures in the complex plane. Let $\beta \in \mathbb Z[\mathrm{i}]$, whose complex conjugate $\overline{\beta}$ is not a divisor of β, and $T \subset \mathbb Z[\mathrm{i}]$ a finite subset. Let µ be a non-atomic self-similar measure with respect to the IFS $\big\{f_{t}(z)=\frac{z+t}{\beta}\colon t\in T\big\}$. For $\alpha \in \mathbb Z[\mathrm{i}]$, if α and β are relatively prime, then we show that the sequence $(\alpha^n z)_{n\ge 1}$ is equidistributed modulo one for µ-almost everywhere $z \in \mathbb{C}$. We also discuss normality of radix expansions in Gaussian integer base, and obtain pointwise normality. Our results generalize partially the classical results in the real line to the complex plane.

Let $ K $ be a compact subset of the d-torus invariant under an expanding diagonal endomorphism with s distinct eigenvalues. Suppose the symbolic coding of K satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which statement (A) does not hold but statement (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of statements (A) and (B) for Bedford–McMullen sponges.

We provide upper bounds for the Assouad spectrum $\dim_A^\theta(\mathrm{Gr}({\kern2pt}f))$ of the graph of a real-valued Hölder or Sobolev function f defined on an interval $I \subset \mathbb{R}$. We demonstrate via examples that all of our bounds are sharp. In the setting of Hölder graphs, we further provide a geometric algorithm which takes as input the graph of an $\alpha$-Hölder continuous function satisfying a matching lower oscillation condition with exponent $\alpha$ and returns the graph of a new $\alpha$-Hölder continuous function for which the Assouad $\theta$-spectrum realizes the stated upper bound for all $\theta\in (0,1)$. Examples of functions to which this algorithm applies include the continuous nowhere differentiable functions of Weierstrass and Takagi.

We define a spherical and hyperbolic analog to the Euclidean projection body for star bodies via the gnomonic projection from the unit sphere and stereographic projection in the hyperbolid model of hyperbolic space. We then prove a spherical and hyperbolic projection inequality for these notions by using an adaption of Steiner symmetrization for spherical, respectively, hyperbolic, star bodies.

Given positive Radon measures, $\mu $ and $\lambda $, on the complex unit circle, we show that absolute continuity of $\mu $ with respect to $\lambda $ is equivalent to their reproducing kernel Hilbert spaces of “analytic Cauchy transforms” in the complex unit disk having dense intersection in the space of $\mu $-Cauchy transforms.

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

Non-autonomous self-similar sets are a family of compact sets which are, in some sense, highly homogeneous in space but highly inhomogeneous in scale. The main purpose of this paper is to clarify various regularity properties and separation conditions relevant for the fine local scaling properties of these sets. A simple application of our results is a precise formula for the Assouad dimension of non-autonomous self-similar sets in $\mathbb{R}^d$ satisfying a certain “bounded neighbourhood” condition, which generalises earlier work of Li–Li–Miao–Xi and Olson–Robinson–Sharples. We also see that the bounded neighbourhood assumption is, in few different senses, as general as possible.

Let $\mu _{M,D}$ be the self-similar measure generated by $M=RN^q$ and the product-form digit set $D=\{0,1,\ldots ,N-1\}\oplus N^{p_1}\{0,1,\ldots ,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\ldots ,N-1\}$, where $R\geq 2$, $N\geq 2$, q, $p_i(1\leq i\leq s)$ are integers with $\gcd (R,N)=1$ and $1\leq p_1<p_2<\cdots <p_s<q$. In this paper, we first show that $\mu _{M,D}$ is a spectral measure with a model spectrum $\Lambda $. Then, we completely settle two types of spectral eigenvalue problems for $\mu _{M,D}$. In the first case, for a real t, we give a necessary and sufficient condition under which $t\Lambda $ is also a spectrum of $\mu _{M,D}$. In the second case, we characterize all possible real numbers t such that $\Lambda '\subset \mathbb {R}$ and $t\Lambda '$ are both spectra of $\mu _{M,D}$.

Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form

$$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$

where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$, $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$, we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$.

We show that the Hausdorff dimension of the attractor of an inhomogeneous self-similar iterated function system (or self-similar IFS) can be well approximated by the Hausdorff dimension of the attractor of another inhomogeneous self-similar IFS satisfying the strong separation condition. We also determine a formula for the Hausdorff dimension of the algebraic product and sum of the inhomogeneous attractor.

In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math. Soc. 146(5) (2018), 1833–1844] made a seminal contribution by linking the improvability of Dirichlet’s theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B> 1$. We determine the Hausdorff dimension of the following set: $ \{x\in [0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text { infinitely often}\}. $

We show that $\alpha $-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for $0<\alpha <1$ and every $\alpha $-stable Lévy motion ${\mathbb {W}}$, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1\leq \alpha <2$ and every symmetric $\alpha $-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1< \alpha <2$ and every $-1\leq \beta \leq 1$ there exists a function f whose associated time series is in the classical domain of attraction of an $S_\alpha (\ln (2), \beta ,0)$ random variable.

We study a family of Thompson-like groups built as rearrangement groups of fractals introduced by Belk and Forrest in 2019, each acting on a Ważewski dendrite. Each of these is a finitely generated group that is dense in the full group of homeomorphisms of the dendrite (studied by Monod and Duchesne in 2019) and has infinite-index finitely generated simple commutator subgroup, with a single possible exception. More properties are discussed, including finite subgroups, the conjugacy problem, invariable generation and existence of free subgroups. We discuss many possible generalisations, among which we find the Airplane rearrangement group $T_A$. Despite close connections with Thompson’s group F, dendrite rearrangement groups seem to share many features with Thompson’s group V.

We prove the following restricted projection theorem. Let $n\ge 3$ and $\Sigma \subset S^{n}$ be an $(n-1)$-dimensional $C^2$ manifold such that $\Sigma$ has sectional curvature $\gt1$. Let $Z \subset \mathbb{R}^{n+1}$ be analytic and let $0 \lt s \lt \min\{\dim Z, 1\}$. Then

\begin{equation*} \dim \{z \in \Sigma\; :\; \dim (Z \cdot z) \lt s\} \le (n-2)+s = (n-1) + (s-1) \lt n-1.\end{equation*}

$z \in \Sigma$

$\dim (Z \cdot z) = \min\{\dim Z, 1\}$

The core idea, originated from Käenmäki–Orponen–Venieri, is to transfer the restricted projection problem to the study of the dimension lower bound of Furstenberg sets of cinematic family contained in $C^2([0,1]^{n-1})$. This cinematic family of functions with multivariables are extensions of those of one variable by Pramanik–Yang–Zahl and Sogge. Since the Furstenberg sets of cinematic family contain the affine Furstenberg sets as a special case, the dimension lower bound of Furstenberg sets improves the one by Héra, Héra–Keleti–Máthé and Dąbrowski–Orponen–Villa.

Moreover, our method to show the restricted projection theorem can also give a new proof for the Mattila projection theorem in $\mathbb{R}^n$ with $n \ge 3$.

We introduce a general class of transport distances $\mathrm {WB}_{\Lambda }$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal {M}(\Omega, \mathbb {S}_+^n)$, called the weighted Wasserstein–Bures distance. Such a distance is defined via a generalised Benamou–Brenier formulation with a weighted action functional and an abstract matricial continuity equation, which leads to a convex optimisation problem. Some recently proposed models, including the Kantorovich–Bures distance and the Wasserstein–Fisher–Rao distance, can naturally fit into ours. We give a complete characterisation of the minimiser and explore the topological and geometrical properties of the space $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$. In particular, we show that $(\mathcal {M}(\Omega, \mathbb {S}_+^n),\mathrm {WB}_{\Lambda })$ is a complete geodesic space and exhibits a conic structure.

We provide two methods to characterise the connectedness of all d-dimensional generalised Sierpiński sponges whose corresponding iterated function systems (IFSs) are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in ${\mathbb {R}}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.