Inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems

TL;DR

This paper studies the size and structure of sets related to inhomogeneous and simultaneous Diophantine approximation within beta dynamical systems, determining their Lebesgue measure and Hausdorff dimension under various conditions.

Contribution

It provides new results on the measure and Hausdorff dimension of approximation sets in beta dynamical systems, extending understanding of Diophantine approximation in these systems.

Findings

Determined Lebesgue measure and Hausdorff dimension of specific approximation sets.

Extended results to simultaneous approximation with variable exponents.

Analyzed approximation sets involving Lipschitz functions and additional assumptions.

Abstract

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $\beta>1$ let $T_{\beta}$ be the $\beta$-transformation on $[0,1]$. We determine the Lebesgue measure and Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: |T_{\beta}^nx-f(x,y)|<\varphi(n)\text{ for infinitely many }n\in\mathbb{N}\right\},\] where $f:[0,1]^2\to [0,1]$ is a Lipschitz function and $\varphi$ is a positive function on $\mathbb{N}$. Let $\beta_2\geq \beta_1>1$, $f_1,f_2:[0,1]\to [0,1]$ be two Lipschitz functions, $\tau_1,\tau_2$ be two positive continuous functions on $[0,1]$. We also determine the Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{\beta_1}^nx-f_1(x)|<\beta_1^{-n\tau_1(x)}\\ &|T_{\beta_2}^ny-f_2(y)|<\beta_2^{-n\tau_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}.\] Under certain additional assumptions, the Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: \begin{aligned}&|T_{\beta_1}^nx-g_1(x,y)|<\beta_1^{-n\tau_1(x)}\\ &|T_{\beta_2}^ny-g_2(x,y)|<\beta_2^{-n\tau_2(y)}\end{aligned}\text{ for infinitely many }n\in\mathbb{N}\right\}\] is also determined, where $g_1,g_2:[0,1]^2\to [0,1]$ are two Lipschitz functions.