Online Deposited in DRO

Let ‖ · ‖ denote the distance to the nearest integer and, for a prime number p, let | · |p denote the p-adic absolute value. In 2004, de Mathan and Teulié asked whether infq≥1 q · ‖qα‖ · |q|p = 0 holds for every badly approximable real number α and every prime number p. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number α grows too rapidly or too slowly, then their conjecture is true for the pair (α, p) with p an arbitrary prime.