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

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.