On the complexity of a putative counterexample to the $p$-adic Littlewood conjecture

TL;DR

This paper investigates the complexity conditions of real numbers' continued fraction expansions that influence the validity of a $p$-adic Littlewood conjecture, providing new cases where the conjecture holds.

Contribution

It establishes that certain growth rates of partial quotient complexities ensure the conjecture's truth for any prime $p$, advancing understanding of the conjecture's scope.

Findings

Conjecture holds if partial quotient complexity grows too rapidly.

Conjecture holds if partial quotient complexity grows too slowly.

Results apply to arbitrary primes $p$.

Abstract

Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teuli\'e asked whether $\inf_{q \ge 1} \, q \cdot || q \alpha || \cdot | q |_p = 0$ holds for every badly approximable real number $\alpha$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number $\alpha$ grows too rapidly or too slowly, then their conjecture is true for the pair $(\alpha, p)$ with $p$ an arbitrary prime.