Bugeaud Collection of Conjectures and Open Questions: $p$-adic Littlewood Conjecture

Problem 10.8 ($p$-adic Littlewood conjecture). For every real number $\xi$ and every prime number $p$, $\inf_{q \ge 1} q \cdot \lVert q \xi \rVert \cdot |q|_p = 0,$ where $\lVert \cdot \rVert$ denotes the distance to the nearest integer and $|\cdot|_p$ denotes the $p$-adic absolute value. Posed by de Mathan and Teulié [dMT04].