Algebraic integers with conjugates in a prescribed distribution

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