  Given a compact subset $\Sigma$ of the real numbers 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 $tr(\alpha) < 1.89831\cdot deg(\alpha)$. We also show how this work can be
applied to find simple abelian varieties over finite fields with extreme point
counts.
