Approximation orders of real numbers by $\beta$-expansions

TL;DR

This paper investigates how well real numbers can be approximated by their $eta$-expansions, establishing approximation rates, Hausdorff dimensions of special sets, and applications to dynamical systems and Diophantine approximation.

Contribution

It proves almost all real numbers are approximated at an exponential rate by their $eta$-expansion convergents and determines the Hausdorff dimensions of sets with different approximation orders.

Findings

Almost all real numbers are approximated with exponential order $eta^{-n}$.

Hausdorff dimensions of sets with various approximation orders are explicitly determined.

Applications include orbit analysis under $eta$-transformation and Diophantine approximation.

Abstract

We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their $\beta$-expansions with the exponential order $\beta^{-n}$. Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under $\beta$-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of $\beta$-expansions.