Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors

Abstract

For every vector $\overline α\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\hat{I}\pm -\overline{p}$. We focus on algebraic vectors, namely $\overline{\hat{I}}\pm =(\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_n)$ such that $1,\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_n$ 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 $\overline{\hat{I}}\pm$ 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 (α_1, \dots, α_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 $\epsilon$-approximations of $(\hat{I}{\pm }_1,\dots ,\hat{I}{\pm }_n)$. Each such limiting measure is expressed as the pushforward of an algebraic measure on $X_n$ to the sphere.