Dynamical Diophantine Approximation

TL;DR

This paper investigates the covering properties of intervals generated by Gibbs measures under the doubling map, linking dynamical systems, multifractal analysis, and inhomogeneous Diophantine approximation, and extends mass transference principles.

Contribution

It introduces a mass transference principle for multifractal Gibbs measures and characterizes the combinatorial structure of typical sequences in this context.

Findings

Established a mass transference principle for multifractal Gibbs measures.

Described the combinatorial structure of typical sequences and occurrence of atypical words.

Connected dynamical properties to inhomogeneous diadic Diophantine approximation.

Abstract

Let $\mu$ be a Gibbs measure of the doubling map $T$ of the circle. For a $\mu$-generic point $x$ and a given sequence $\{r_n\} \subset \R^+$, consider the intervals $(T^nx - r_n \pmod 1, T^nx + r_n \pmod 1)$. In analogy to the classical Dvoretzky covering of the circle we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hitting times for moving targets. A mass transference principle is obtained for Gibbs measures which are multifractal. Such a principle was shown by Beresnevich and Velani \cite{BV} only for monofractal measures. In the symbolic language we completely describe the combinatorial structure of a typical relatively short sequence, in particular we can describe the occurrence of ''atypical'' relatively long words. Our results have a direct and deep number-theoretical interpretation via inhomogeneous diadic diophantine approximation by numbers belonging to a given (diadic) diophantine class.