Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors

Abstract

For every vector $\overline \alpha\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\alpha-\overline p$. We focus on algebraic vectors, namely $\overline \alpha=(\alpha_1,\dots,\alpha_n)$ such that $1, \alpha_1, \dots, \alpha_n$ span a rank $n$ number field. For these vectors, we investigate the size of their displacements as well as the distribution of their directions. We give a new proof to the result of Bugeaud in \cite{YannPAdic} saying that algebraic vectors $\overline \alpha$ satisfy the $p$-adic Littlewood Conjecture. Namely, we prove that \begin{equation} \liminf_{k \to \infty} \left( k \abs{k}_p \right)^{1/n} \| k (\alpha_1, \dots, \alpha_n) \|_\infty = 0. \end{equation} Our new proof lets us classify all limiting distributions, with a special weighting, of the sequence of directions of the defects in the $\varepsilon$-approximations of $(\alpha_1, \dots, \alpha_n)$. Each such limiting measure is expressed as the pushforward of an algebraic measure on $X_n$ to the sphere.