On Uniform Distribution of Algebraic Numbers

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 t...

On the Approximation to Algebraic Numbers by Algebraic Numbers

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...

Notes on Algebraic Numbers

This is a summary of my 1994–1995 course on Algebraic Numbers. (Revised and improved on 1993– 1994!) The background assumed is standard elementary number theory—as found in my Level III course—and a little (Abelian) group theory. Corrections and suggestions for improvement are welcome, and will be credited in future editions! I first learned algebraic number theory from Stewart & Tall’s book ([...

Record Range of Uniform Distribution

We consider a sequence of independent and identicaly distributed (iid) random variables with absolutely continuous distribution function F(x) and probability density function (pdf) f(x). Let Rnl be the largest observation after observing nth record and R(ns) be the smallest observation after observing the nth record. Then we say Wnr = Rnl− R(ns), n > 1, as the nth record range. We will c...

On approximation of real numbers by real algebraic numbers