On the rational approximation to Thue--Morse rational 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 $\exp (K(\log b{)}^2\sqrt{\ell }{2}^{\ell /2})$.