On the rational approximation to Thue--Morse rational numbers

TL;DR

This paper investigates the approximation properties of specific Thue--Morse related rational numbers, establishing bounds on their partial quotients that depend on parameters like base and level, revealing their Diophantine approximation characteristics.

Contribution

The paper provides explicit bounds on the partial quotients of a class of Thue--Morse rational numbers, linking their structure to Diophantine approximation theory.

Findings

Bound on partial quotients depending on base and level

Explicit formula for denominator of the rational numbers

Insight into the approximation complexity of Thue--Morse related numbers

Abstract

Let $b \ge 2$ and $\ell \ge 1$ be integers. We establish that there is an absolute real number $K$ such that all the partial quotients of the rational number $$ \prod_{h = 0}^\ell \, (1 - b^{-2^h}), $$ of denominator $b^{2^{\ell+1} - 1}$, do not exceed $\exp(K (\log b)^2 \sqrt{\ell} 2^{\ell/2})$.