Diophantine analysis of the expansions of a fixed point under continuum many bases

TL;DR

This paper investigates the Diophantine approximation properties of orbits of points under beta-transformations across a continuum of bases, establishing measure-theoretic results related to approximation rates and extending prior work significantly.

Contribution

It provides a comprehensive analysis of approximation behaviors for orbits under varying bases, strengthening existing results and introducing methods to determine measure properties of approximation sets.

Findings

Almost all bases allow orbits to approximate sequences based on series divergence.

The measure of points approximated infinitely often is characterized for fixed bases.

The approach extends to approximation involving Lipschitz functions and Lebesgue measure.

Abstract

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 sequence of real numbers in $[0,1]$ and $\varphi\colon \mathbb{N}\rightarrow (0,1]$ be a positive function. With a detailed analysis on the distribution of {\em full cylinders} in the base space $\{\beta>1\}$, it is shown that for any given $x\in(0,1]$, for almost all or almost no bases $\beta>1$, the orbit of $x$ under $T_{\beta}$ can $\varphi$-well approximate the sequence $\{x_{n}\}_{n\geq 1}$ according to the divergence or convergence of the series $\sum \varphi(n)$. This strengthens Schmeling's result significantly and complete all known results in this aspect. Moreover, the idea presented here can also be used to determine the Lebesgue measure of the set \begin{equation*} \{x\in [0,1]\colon|T^{n}_{\beta}x-L(x)|<\varphi(n) \text{ for infinitely many } n\in\mathbb{N}\}, \end{equation*} for a fixed base $\beta>1$, where $L\colon [0,1]\rightarrow[0,1]$ is a Lipschitz function.