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

Abstract

Let $p$ be a prime number and $ξ$ an irrational $p$-adic number. Its multiplicative irrationality exponent ${μ^{\times}} (ξ)$ is the supremum of the real numbers ${μ^{\times}}$ for which the inequality $$ |b ξ- a|_{p} \leq | a b |^{- {μ^{\times}} / 2} $$ has infinitely many solutions in nonzero integers $a, b$. We show that ${μ^{\times}} (ξ)$ can be expressed in terms of a new exponent of approximation attached to a sequence of rational numbers defined in terms of $ξ$. We establish that ${μ^{\times}} ({{ξ_{{\bf t}, p}}}) = 3$, where ${{ξ_{{\bf t}, p}}}$ 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 $\{-1, 1\}$.