A p-adic criterion for Lehmer's conjecture

For a non-zero algebraic number $\alpha$ of degree $d$, let $h(\alpha )$ denote its logarithmic Weil height. It is known that when $h(\alpha )$ is small, and $d$ is large, the conjugates of $\alpha$ are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a $p$-adic analogue of this result by obtaining lower bounds for $h(\alpha )$ in terms of the number of its conjugates that lie in a finite extension of ${\mathbb{Q}}_p$, for some prime $p$. As a consequence, we prove Lehmer’s conjecture for all $\alpha$ such that $\gg \sqrt{d\log d}$ many of its conjugates lie in a finite extension of ${\mathbb{Q}}_p$.