Galois Distribution on Tori—A Refinement, Examples, and Applications

Abstract In this paper we give a refinement of the one-dimensional form of Bilu’s Equidistribution Theorem together with some examples and applications. These include an explicit bound for the sum of the $e^{i\theta }$ taken over all conjugates $re^{i\theta }$ of an algebraic number $\alpha $ in terms of its height $h$ and degree $d$. That is applied to show that a certain discrepancy of $\alpha $ is at most $8000h^{1/3}$, which removes an extra term involving $d$ from previous bounds; this cannot be done simply by using a lower bound for $h$ in terms of $d$, even assuming the Lehmer Conjecture. And we give a new estimate for the norm of $1-\alpha $. We also improve existing upper bounds for the height of $\xi $ when $\xi ,1-\xi $ are multiplicatively dependent.