Distribution of algebraic numbers

TL;DR

This paper investigates the distribution of algebraic numbers, extending classical results to complex sets, providing new bounds, and introducing the generalized Mahler measure to characterize equidistribution.

Contribution

It generalizes the Erdős-Turán theorem, solves a problem on algebraic numbers on the real line, and introduces the generalized Mahler measure for distribution analysis.

Findings

Algebraic numbers in the unit disk have constrained growth.

The rate of convergence of arithmetic means is estimated.

Generalized Mahler measure characterizes algebraic number distribution.

Abstract

Schur studied limits of the arithmetic means $A_n$ of zeros for polynomials of degree $n$ with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that $\limsup_{n\to\infty} |A_n| \le 1-\sqrt{e}/2.$ We show that $A_n \to 0$, and estimate the rate of convergence by generalizing the Erd\H{o}s-Tur\'an theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for $\liminf_{n\to\infty} A_n$ was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line. Potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure.