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arXiv · 2304.10021

A quantitative converse of Fekete's theorem

Naser Talebizadeh Sardari·Bryce Joseph Orloski
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Abstract

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.

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Naser Talebizadeh Sardari, Bryce Joseph Orloski. 2023-04-20. A quantitative converse of Fekete's theorem. https://arxiv.org/abs/2304.10021

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