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

Abstract

Let denote the distance to the nearest integer and, for a prime number , let denote the -adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229-245] asked whether holds for every badly approximable real number and every prime number . Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number grows too rapidly or too slowly, then their conjecture is true for the pair with an arbitrary prime.