Estimating lower limit in the $p$-adic Littlewood conjecture

Abstract

We verify that $\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx||<\epsilon$ for all real $x$, small primes $p$ and relatively small $\epsilon$. This result supports the famous $p$-adic Littlewood conjecture which states that the above lower limit is equal to 0 for all $x\in\mathbb{R}$. In particular, the result is established for $p=2$ with $\epsilon=1/25$. For $3\le p\le 29$, the upper bounds for $\epsilon$ vary, but they are always at most $1/10$.