On the rational approximation to $p$-adic Thue–Morse numbers

Abstract

Let $p$ be a prime number and let $\xi$ be an irrational $p$-adic number. Its multiplicative irrationality exponent $\mu^{\times}(\xi)$ is the supremum of the real numbers $\mu^{\times}$ for which the inequality $|b \xi - a|{p} \leq | a b |^{-\mu^{\times} / 2}$ has infinitely many solutions in nonzero integers $a$ and $b$. We show that $\mu^{\times} (\xi)$ can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of $\xi$. We establish that $\mu^{\times} ({\xi{\mathbf{t}, p}}) = 3$, where ${\xi\_{\mathbf{t}, p}}$ is the $p$-adic number $1 - p - p^2 + p^3 - p^4 + \ldots$, whose sequence of digits is given by the Thue–Morse sequence over ${-1, 1}$.