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

TL;DR

This paper investigates the multiplicative irrationality exponent of certain $p$-adic numbers, establishing a precise value for a number constructed from the Thue--Morse sequence, and introduces a new approximation exponent linked to rational sequences.

Contribution

It introduces a new exponent of approximation for $p$-adic numbers and computes its value for a specific Thue--Morse based $p$-adic number.

Findings

The multiplicative irrationality exponent of the $p$-adic Thue--Morse number is exactly 3.

A new approximation exponent related to rational sequences is defined and analyzed.

The paper connects $p$-adic approximation properties with combinatorial sequences like Thue--Morse.

Abstract

Let $p$ be a prime number and $\xi$ 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 $a, b$. 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 ${{\mu^{\times}}} ({{\xi_{{\bf t}, p}}}) = 3$, where ${{\xi_{{\bf t}, p}}}$ 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 $\{-1, 1\}$.