  Given a compact subset $\Sigma \subset \mathbb{R}$ (or $\mathbb{C}$) with
logarithmic capacity greater than zero, we construct an explicit family of
probability measures supported on $\Sigma$ such that their closure is all the
possible weak limit measures of complete sets of conjugate algebraic integers
lying inside $\Sigma$. We give an asymptotic formula for the number of
algebraic integers with given degree and prescribed distribution. We exploit
the algorithmic nature of our approach to give a family of upper bounds that
converges to the smallest limiting trace-to-degree ratio of totally positive
algebraic integers and improve the best previously known upper bound on the
Schur-Siegel-Smyth trace problem to 1.8216.
