Related papers: Dispersion and Littlewood's conjecture

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…

Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{\alpha\in\mathcal{A}\cap[1,x]}\frac{1}{\alpha}>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many…

For every vector $\overline \alpha\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\alpha-\overline p$. We focus on algebraic vectors, namely $\overline…

We consider a natural filtration $\boldsymbol{\operatorname{Bad}}(\delta) \subset \boldsymbol{\operatorname{Bad}}(\delta')$ for $\delta \geq \delta'>0$ on the set of badly approximable numbers to complement the filtration of the well…

Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$…

Let $\beta>1$ be a non-integer. First we show that Lebesgue almost every number has a $\beta$-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many $\beta$-expansions of the same given frequency.…

We study problems on covering $[0,1)$ by shrinking intervals centered at the points $\{q_n x\}$, where $(q_n)_{n\in \mathbb{N}}$ is a given real-valued sequence and $x \in [0,1)$ is random. For real-valued lacunary sequences…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

Given a strictly increasing sequence of positive real numbers tending to infinity $(q_{n})_{n=1}^{\infty}$, and an arbitrary sequence of real numbers $(r_{n})_{n=1}^{\infty}.$ We study the set of $\alpha\in(1,\infty)$ for which…

In this paper, we study the Diophantine properties of the orbits of a fixed point in its expansions under continuum many bases. More precisely, let $T_{\beta}$ be the beta-transformation with base $\beta>1$, $\{x_{n}\}_{n\geq 1}$ be a…

We prove that the Littlewood conjecture is satisfied for a restricted class of pairs $(\alpha,\beta)$ of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine…

By means of Peres-Schlag's method we prove the existence of real numbers $\alpha, \beta$ such that $$ \liminf_{q\to \infty} (q\log^2 q)||\alpha q|| ||\beta q|| > 0.

We establish a strong form of Littlewood's conjecture with inhomogeneous shifts, for a full-dimensional set of pairs of badly approximable numbers on a vertical line. We also prove a uniform assertion of this nature, generalising a strong…

The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $\alpha>0$, and an infinite set $\Gamma\subseteq…

Suppose $\alpha, \beta$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $\gamma$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $\alpha(0) = \gamma(0) = 0$, and $\alpha(1) = \beta(0) = 0$ and…

Given a real number $\alpha \in (0,1)$, we define the Webster sequence of density $\alpha$ to be $W_\alpha = (\lceil(n-1/2) / \alpha\rceil)_{n\in\mathbb{N}}$, where $\lceil x \rceil$ is the ceiling function. It is known that if $\alpha$ and…

The coefficients $A_n(\alpha,\beta,\omega)$ in the Maclaurin expansion $(1+\omega z)^{\alpha}(1-z)^{-\beta}= \sum_{n=0}^{\infty} A_n(\alpha,\beta,\omega)z^n$ are studied, where $\omega,z \in \mathbb{C}$ with $|z| < |\omega|=1$, and…

We prove an inequality related to questions in Approximation Theory, Probability Theory, and to Irregularities of Distribution. Let $h_R$ denote an $L ^{\infty}$ normalized Haar function adapted to a dyadic rectangle $R\subset [0,1] ^{3}$.…

Let the sequence $\{t_n\}_{n=1}^{\infty}$ of reals satisfy the condition $ \frac{t_{n+1}}{t_n} \ge 1+ \frac{\gamma}{n^\beta},0\le \beta <1, \gamma >0. $ Then the set $ \{\alpha \in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||t_n…

Let $\beta\in(1,2)$ and $x\in [0,\frac{1}{\beta-1}]$. We call a sequence $(\epsilon_{i})_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ a $\beta$-expansion for $x$ if $x=\sum_{i=1}^{\infty}\epsilon_{i}\beta^{-i}$. We call a finite sequence…