Related papers: Prime-representing functions and Hausdorff dimensi…

Let $X=\bigcup\varphi_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $\varphi_{i}$, we…

The set of badly approximable numbers, Bad, is known to be winning for Schmidt's game and hence has full Hausdorff dimension. It is also known that the set of inhomogeneously badly approximable numbers has full dimension. We prove that the…

Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points…

Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. For maps like multiplication by an integer modulo 1, such sets have full…

For a decreasing real valued function $\psi$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $\psi$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m <…

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to…

The Jarn\'ik-Besicovitch theorem is a fundamental result in metric number theory which concerns the Hausdorff dimension for certain limsup sets. We discuss the analogous problem for liminf sets. Consider an infinite sequence of positive…

We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in $\mathbb{R}^n$. As an application, we find an upper bound close to $n-k$ for the Hausdorff dimension of sets with large $k$-porosity. With $k$-porous sets we…

We construct a subring as mentioned in the title (hence this subring has Lebesgue measure zero).

Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each…

Let us consider a sphere $S^{n-1}$ of radius $r$ in $\mathbb{R}^n$, where we have fixed poles $N$ and $S$. Suppose that $K$ is a set in $\mathbb{R}^n$ containing a translated copy of each meridian (that is an $S^{n-2}$-sphere) of $S^{n-1}$.…

Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density condition and ${\bf…

Given an integer $N\ge 2$ and a real number ${\beta}>1$, let $\Gamma_{{\beta},N}$ be the set of all $x=\sum_{i=1}^\infty {d_i}/{{\beta}^i}$ with $d_i\in\{0,1,\cdots,N-1\}$ for all $i\ge 1$. The infinite sequence $(d_i)$ is called a…

This paper investigates the Hausdorff measure of certain sets of generics in computability theory. Let $\Gamma$ be the Turing ideal in which we take the dense open sets. The set of $\Gamma$-Cohen generics has measure positive if and only if…

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\dots\le|\lambda_d|$. We investigate the Hausdorff dimension of the…

We observe that a probability distribution supported by $\mathbb{N}$, induces a representation of real numbers in [0, 1) with digits in $\mathbb{N}$. We first study the Hausdorff dimension of sets with prescribed digits with respect to…

We introduce a novel deterministic fractal set PF in the unit interval whose construction is driven by the sequence of prime numbers modulo 16. At each step of the recursive construction, two subintervals are retained based on the residues…

We consider digits-deleted sets or Cantor-type sets with $\beta$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $\beta$. The $d$-dimentional Hausdorff measure of these sets…

A cuf space (set, resp.) is a space (set, resp.) which is a countable union of finite subspaces (subsets, resp.). It is proved in $\mathbf{ZF}$ (with the absence of the axiom of choice) that all countable unions of cuf (denumerable, resp.)…

We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.