POLYNOMIALS WITH ZEROS ON THE UNIT CIRCLE: REGULARITY OF LEJA SEQUENCES

Abstract

Let z 1 , ⋯ , z m be m distinct complex numbers, normalized to | z k | = 1 , and consider the polynomial p m ( z ) = ∏ k = 1 m ( z − z k ) . We define a sequence of polynomials in a greedy fashion, p N + 1 ( z ) = p N ( z ) z − z ∗ where z ∗ = arg max | z | = 1 | p N ( z ) | , and prove that, independently of the initial polynomial p m , the roots of p N equidistribute in angle at rate at most ( log N ) 2 / N . This persists when sometimes adding “adversarial” points by hand. We also obtain sharp rates for an L2-version of a problem first raised by Erdős and solved by Beck in L ∞ .