A quantitative converse of Fekete's theorem

Given a compact subset $\mathrm{\Sigma }\subset \mathbb{R}$ (or $\mathbb{C}$) with logarithmic capacity greater than zero, we construct an explicit family of probability measures supported on $\mathrm{\Sigma }$ such that their closure is all the possible weak limit measures of complete sets of conjugate algebraic integers lying inside $\mathrm{\Sigma }$. 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.