Directional $p$-Adic Littlewood Conjecture for Algebraic Vectors

For every vector $\bar{\alpha }\in {\text{RR}}^n$ and for every rational approximation $(\bar{p},q)\in {\text{RR}}^n\times \text{RR}$ we can associate the displacement vector $q\alpha -\bar{p}$. We focus on algebraic vectors, namely $\bar{\alpha }=({\alpha }_1,\dots ,{\alpha }_n)$ 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 $\bar{\alpha }$ 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 (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 $({\alpha }_1,\dots ,{\alpha }_n)$. Each such limiting measure is expressed as the pushforward of an algebraic measure on $X_n$ to the sphere.