On the complexity of a putative counterexample to the -adic Littlewood conjecture
Dmitry Badziahin, Yann Bugeaud, Manfred Einsiedler, Dmitry Kleinbock · Compositio Mathematica · 2015
Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$ , let $|\cdot |_{p}$ denote the $p$ -adic absolute value. Over a decade ago, de Mathan and Teulié [ Problèmes diophantiens simultanés , Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$ $q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$ . Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.