Algebraic integers with conjugates in a prescribed distribution

Abstract

Given a compact subset $\Sigma$ of $\mathbb{R}$ obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $\Sigma$. The distribution of conjugates of such an integer defines a probability measure on $\Sigma$; our main result gives a necessary and sufficient condition for a given probability measure on $\Sigma$ to be the limit of some sequence of distributions of conjugates. As one consequence, we show there are infinitely many totally positive algebraic integers $\alpha$ with $\mathrm{tr}(\alpha)\lt 1.89831 \cdot \mathrm{deg}(\alpha)$. We also show how this work can be applied to find simple abelian varieties over finite fields with extreme point counts.