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 $|b\xi -a{|}_p\le |ab{|}^{-{\mu }^{\times }/2}$ 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 $1-p-p^2+p^3-p^4+\dots$, whose sequence of digits is given by the Thue–Morse sequence over $\{-1,1\}$.