Distribution of Algebraic Numbers

Schur studied limits of the arithmetic means An 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 lim supn→∞ |An| ≤ 1 − √ e/2. We show that An → 0, and estimate the rate of convergence by generalizing the Erdős-Turán 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 lim infn→∞An 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. 1. Schur’s problems on means of algebraic numbers Let E be a subset of the complex plane C. Consider the set of polynomials Zn(E) of the exact degree n with integer coefficients and all zeros in E. We denote the subset of Zn(E) with simple zeros by Zn(E). Given M > 0, we write Pn = anz+ . . . ∈ Zn(E,M) if |an| ≤M and Pn ∈ Zn(E) (respectively Pn ∈ Zn(E,M) if |an| ≤M and Pn ∈ Zn(E)). Schur [45], §4-8, studied the limit behavior of the arithmetic means of zeros for polynomials from Zn(E,M) as n → ∞, where M > 0 is an arbitrary fixed number. His results may be summarized in the following statements. Let R+ := [0,∞), where R is the real line. Theorem A (Schur [45], Satz IX) Given a polynomial Pn(z) = an ∏n k=1(z − αk,n), define the arithmetic mean of squares of its zeros by Sn := ∑n k=1 α 2 k,n/n. If Pn ∈ Zn(R,M) is any sequence of polynomials with degrees n→∞, then lim inf n→∞ Sn ≥ √ e > 1.6487. (1.1) Theorem B (Schur [45], Satz XI) For a polynomial Pn(z) = an ∏n k=1(z − αk,n), define the arithmetic mean of its zeros by An := ∑n k=1 αk,n/n. If Pn ∈ Zn(R+,M) is any sequence of polynomials with degrees n→∞, then lim inf n→∞ An ≥ √ e > 1.6487. (1.2) 1991 Mathematics Subject Classification. Primary 11C08; Secondary 11R04, 26C10, 30C15.