One more proof of the Erd\H{o}s--Tur\'an inequality, and an error estimate in Wigner's law

TL;DR

This paper proves the Erdős–Turán inequality using classical approximation methods, extends its applicability, and provides a sharp error estimate in Wigner's law near the spectral edge.

Contribution

It introduces a new proof of the Erdős–Turán inequality via Chebyshev, Markov, and Stieltjes methods, and applies this to derive a sharp edge error estimate in Wigner's law.

Findings

Proved Erdős–Turán inequality with a classical approach.

Extended the inequality's applicability to more general settings.

Provided a sharp error estimate in Wigner's law near the spectral edge.

Abstract

We prove the Erdos-Turan equidistribution inequality, using a construction due to Chebyshev, Markov, and Stieltjes. The method is applicable in a more general setting. As an example, we state another inequality that can be proved using this method. The inequality can be used to get an error estimate in Wigner's law which is sharp near the edge.