The Sharp Erd\H{o}s-Tur\'an Inequality

TL;DR

This paper establishes the sharp form of the Erdős-Turán inequality by reformulating it as an energy minimization problem and analyzing the equilibrium distributions using potential theory and complex analysis.

Contribution

It provides the exact optimal constant in the Erdős-Turán inequality, solving a longstanding open problem in polynomial root distribution.

Findings

Derived the explicit form of the equilibrium distributions

Proved the inequality with the optimal constant

Connected the problem to energy minimization and potential theory

Abstract

Erd\H{o}s and Tur\'an proved a classical inequality on the distribution of roots for a complex polynomial in 1950, depicting the fundamental interplay between the size of the coefficients of a polynomial and the distribution of its roots on the complex plane. Various results have been dedicated to improving the constant in this inequality, while the optimal constant remains open. In this paper, we give the optimal constant, i.e., prove the sharp Erd\H{o}s-Tur\'an inequality. To achieve this goal, we reformulate the inequality into an optimization problem, whose equilibriums coincide with a class of energy minimizers with the logarithmic interaction and external potentials. This allows us to study their properties by taking advantage of the recent development of energy minimization and potential theory, and to give explicit constructions via complex analysis. Finally the sharp Erd\H{o}s-Tur\'an inequality is obtained based on a thorough understanding of these equilibrium distributions.