Related papers: The Duffin-Schaeffer Conjecture with extra diverge…

Let $T_\beta$ be the $\beta$-transformation on $[0,1)$ defined by $$T_\beta(x)=\beta x\text{ mod }1.$$ We study the Diophantine approximation of the orbit of a point $x$ under $T_\beta$. Precisely, for given two positive functions…

Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant…

As a generalization of Dempster-Shafer theory, D number theory provides a framework to deal with uncertain information with non-exclusiveness and incompleteness. However, some basic concepts in D number theory are not well defined. In this…

In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming…

Let $(a_n)_{n \geq 1}$ be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of $\{a_n \alpha \bmod 1, 1 \leq n \leq N\}$ as $N \to \infty$, valid for Lebesgue-almost…

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $\mu$ be a finite Borel measure on $\mathbb{R}^n$ and let $0 < m < n$ be an integer. We show that, under the sole assumption…

Given a compact set of real numbers, a random $C^{m + \alpha}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has…

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ be its continued fraction expansion with partial quotients ${a_n(x), n\ge 1}$. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to \infty$. In this…

We consider badly approximable numbers in the case of dyadic diophantine approximation. For the unit circle $\mathbb{S}$ and the smallest distance to an integer $\|\cdot\|$ we give elementary proofs that the set $F(c) = \{x \in \mathbb{S}:…

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in\mathbb{N},$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} {align*}…

The idea of using measure theoretic concepts to investigate the size of number theoretic sets, originating with E. Borel, has been used for nearly a century. It has led to the development of the theory of metrical Diophantine approximation,…

We show that in $\mathbb{R}^d$ there are purely unrectifiable sets of Hausdorff (and even box counting) dimension $d-1$ which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same…

Besicovitch showed that if a set is null for the Hausdorff measure associated to a given dimension function, then it is still null for the Hausdorff measure corresponding to a smaller dimension function. We prove that this is not true for…

We study de la Vall\'ee Poussin means of Walsh--Fourier series associated with a nondecreasing window sequence. We establish a sharp criterion for almost everywhere convergence for integrable functions. We further show that, when this…

We prove that there exist arbitrarily small positive real numbers $\epsilon$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} \epsilon |\log \vepsilon|^{-1}$ to the set of rational integers. This is…

For a numerical semigroup S $\subseteq$ N with embedding dimension e, conductor c and left part L = S $\cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $\ge$ 0 always holds, a question known…

For a positive integer $n$ and a real number $\xi$, let $\lambda_n (\xi)$ denote the supremum of the real numbers $\lambda$ for which there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi^2 ||, \ldots , ||q…

We show that the set of numbers that are $Q$-distribution normal but not simply $Q$-ratio normal has full Hausdorff dimension. It is further shown under some conditions that countable intersections of sets of this form still have full…

The Dvoretzky--Kiefer--Wolfowitz (DKW) inequality says that if $F_n$ is an empirical distribution function for variables i.i.d.\ with a distribution function $F$, and $K_n$ is the Kolmogorov statistic $\sqrt{n}\sup_x|(F_n-F)(x)|$, then…

For a real analytic periodic function $\phi:\mathbb{R}\to\mathbb{R}^d$, an integer $b \ge 2$ and $\lambda\in(1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function…