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 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.

Let $B^{H}$ be a d-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f\,:\,[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the non-centered Gaussian process $B^{H}+f$. It aims to highlight how each component f, E and F is involved in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$. When F is a singleton and f is a general measurable drift, some new estimates are obtained for the last probability by means of suitable Hausdorff measure and capacity of the graph $Gr_E(f)$. As application we deal with the issue of polarity of points for $(B^H+f)\vert_E$ (the restriction of $B^H+f$ to the subset $E\subset (0,\infty)$).

A classical result of Erdős, Lovász and Spencer from the late 1970s asserts that the dimension of the feasible region of densities of graphs with at most k vertices in large graphs is equal to the number of non-trivial connected graphs with at most k vertices. Indecomposable permutations play the role of connected graphs in the realm of permutations, and Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of permutation patterns of size at most k is at least the number of non-trivial indecomposable permutations of size at most k. However, this lower bound is not tight already for $k=3$. We prove that the dimension of the feasible region of densities of permutation patterns of size at most k is equal to the number of non-trivial Lyndon permutations of size at most k. The proof exploits an interplay between algebra and combinatorics inherent to the study of Lyndon words.

In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine iterated function systems. Also, we study Besicovitch–Eggleston sets for finite generalized Lüroth series number systems with redundancy. The redundancy refers to the fact that each number $x \in [0,1]$ has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.

In this paper,the linear space $\mathcal F$ of a special type of fractal interpolation functions (FIFs) on an interval I is considered. Each FIF in $\mathcal F$ is established from a continuous function on I. We show that, for a finite set of linearly independent continuous functions on I, we get linearly independent FIFs. Then we study a finite-dimensional reproducing kernel Hilbert space (RKHS) $\mathcal F_{\mathcal B}\subset\mathcal F$, and the reproducing kernel $\mathbf k$ for $\mathcal F_{\mathcal B}$ is defined by a basis of $\mathcal F_{\mathcal B}$. For a given data set $\mathcal D=\{(t_k, y_k) : k=0,1,\ldots,N\}$, we apply our results to curve fitting problems of minimizing the regularized empirical error based on functions of the form $f_{\mathcal V}+f_{\mathcal B}$, where $f_{\mathcal V}\in C_{\mathcal V}$ and $f_{\mathcal B}\in \mathcal F_{\mathcal B}$. Here $C_{\mathcal V}$ is another finite-dimensional RKHS of some classes of regular continuous functions with the reproducing kernel $\mathbf k^*$. We show that the solution function can be written in the form $f_{\mathcal V}+f_{\mathcal B}=\sum_{m=0}^N\gamma_m\mathbf k^*_{t_m} +\sum_{j=0}^N \alpha_j\mathbf k_{t_j}$, where ${\mathbf k}_{t_m}^\ast(\cdot)={\mathbf k}^\ast(\cdot,t_m)$ and $\mathbf k_{t_j}(\cdot)=\mathbf k(\cdot,t_j)$, and the coefficients γm and αj can be solved by a system of linear equations.

Let $[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]$ be the continued fraction expansion of $x\in [0,1)$ and $q_n(x)$ be the denominator of its nth convergent. The irrationality exponent and Khintchine exponent of x are respectively defined by

$$ \begin{align*} \overline{v}(x)=2+\limsup_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_n(x)} \quad \text{and}\quad \gamma(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\log a_i(x). \end{align*} $$

We study the multifractal spectrum of the irrationality exponent and the Khintchine exponent for continued fractions with nondecreasing partial quotients. For any $v>2$, we completely determine the Hausdorff dimensions of the sets $\{x\in [0,1): a_1(x)\leq a_2(x)\leq \cdots , \overline {v}(x)=v\}$ and

$$ \begin{align*}\bigg\{x\in[0,1): a_1(x)\leq a_2(x)\leq\cdots, \lim\limits_{n\to\infty}\frac{\log a_1(x)+\log a_2(x)+\cdots+\log a_n(x)}{\psi(n)}=1\bigg\},\end{align*} $$

where $\psi :\mathbb {N}\rightarrow \mathbb {R}^+$ is a function satisfying $\psi (n)\to \infty $ as $n\to \infty $.

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 $

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

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

We study the exact Hausdorff and packing dimensions of the prime Cantor set, $\Lambda _P$, which comprises the irrationals whose continued fraction entries are prime numbers. We prove that the Hausdorff measure of the prime Cantor set cannot be finite and positive with respect to any sufficiently regular dimension function, thus negatively answering a question of Mauldin and Urbański (1999) and Mauldin (2013) for this class of dimension functions. By contrast, under a reasonable number-theoretic conjecture we prove that the packing measure of the conformal measure on the prime Cantor set is in fact positive and finite with respect to the dimension function $\psi (r) = r^\delta \log ^{-2\delta }\log (1/r)$, where $\delta $ is the dimension (conformal, Hausdorff, and packing) of the prime Cantor set.

We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

We define balanced self-similar quasi-round carpets and compare the carpet moduli of some path families relating to such a carpet. Then, using some known results on quasiconformal geometry of carpets, we prove that the group of quasisymmetric self-homeomorphisms of every balanced self-similar quasi-round carpet is finite. Furthermore, we prove that some balanced self-similar carpets in the unit square with strong geometric symmetry are quasisymmetrically rigid by using the quasisymmetry of weak tangents of carpets.

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set

$$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$

$y\in [0,1)$

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 $y_1,\,\ldots,\, y_p \in K_\lambda$.

We show that properties of pairs of finite, positive, and regular Borel measures on the complex unit circle such as domination, absolute continuity, and singularity can be completely described in terms of containment and intersection of their reproducing kernel Hilbert spaces of “Cauchy transforms” in the complex unit disk. This leads to a new construction of the classical Lebesgue decomposition and proof of the Radon–Nikodym theorem using reproducing kernel theory and functional analysis.

We introduce new types of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks [Mar16]. The motivation for the construction comes from the adaptation of this method to the $\mathsf {LOCAL}$ model of distributed computing [BCG+21]. Our approach unifies the previous results in the area, as well as produces new ones. In particular, strengthening the main result of [TV21], we show that for $\Delta>2$, it is impossible to give a simple characterization of acyclic $\Delta $-regular Borel graphs with Borel chromatic number at most $\Delta $: such graphs form a $\mathbf {\Sigma }^1_2$-complete set. This implies a strong failure of Brooks-like theorems in the Borel context.

We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamic formalism where, for dominated and irreducible matrices, we completely characterise the behaviour of the pressures.