The Fully Inhomogeneous $p$-Adic Littlewood Conjecture

Abstract

We prove that for almost every $\alpha\in\RR$, the \textit{fully inhomogeneous $p$-adic Littlewood Conjecture} holds; namely, \begin{equation*} \forall \delta\in\RR,\forall \kappa\in\ZZ_p, \;\;\liminf_{\av{q}\rightarrow+\infty}q\inn{q\alpha+\delta}\abs{q+\kappa}_p=0. \end{equation*} Here, $\inn{\cdot}$ denotes the distance to the nearest integer and $|\cdot|_p$ denotes the $p$-adic norm. Moreover, we prove this conjecture for every quadratic irrational $\alpha$. This gives an affirmative answer to the $p$-adic version of a question posed by Cassels in \cite{Cas59}.