On the p-adic Wirsing problem
For a real transcendental number $ξ$, let $ω_n^*(ξ)$ denote the supremum of all $ω$ for which there exist infinitely many real algebraic numbers $α$ of degree $\leq n$ satisfying $|ξ-α|\leq H(α)^{-ω-1}$, where $H(α)$ is the naive height of the minimal polynomial of $α$. A celebrated result of Wirsing gives the uniform lower bound $ω_n^*(ξ)\geq\frac{n+1}{2}$, which was improved significantly in a recent work of Poëls to $\frac{n}{2-\log 2}$. In this paper, we establish a $p$-adic counterpart of Poëls's result. Let $p$ be a prime and $ξ\in\Qp$ be transcendental. Let $ω_{n,p}^*(ξ)$ be the supremum of all real numbers $ω$ for which there exist infinitely many algebraic numbers $α\in \Qp$ of degree $\leq n$ such that $|ξ-α|_p\leq H(α)^{-ω-1}$. We show that $ω^*_{n,p}(ξ)\geq\frac{n}{2-\log 2}-1$. This improves the known lower bounds in the $p$-adic setting, namely the analogue of Wirsing's theorem, due to Morrison and Teulié.