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

TL;DR

This paper provides partial verification of the $p$-adic Littlewood conjecture by establishing a positive lower limit for the product involving $q$, the $p$-adic absolute value, and the distance to the nearest integer, for small primes and small epsilon.

Contribution

It offers the first bounds supporting the conjecture for small primes and specific epsilon values, advancing understanding of the conjecture's validity.

Findings

Verified the lower limit for $p=2$ with $oxed{1/25}$

Established bounds for $3 \\le p \\le 29$, always at most $1/10$

Supports the conjecture that the limit is zero for all real $x$

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 $1/10$.