ROOTS OF TRIGONOMETRIC POLYNOMIALS AND THE ERDŐS–TURÁN THEOREM

Abstract

We prove, informally put, that it is not a coincidence that cos ( n θ ) + 1 ≥ 0 and the roots of z n + 1 = 0 are uniformly distributed in angle—a version of the statement holds for all trigonometric polynomials with “few” real roots. The Erdős–Turán theorem states that if p ( z ) = ∑ k = 0 n a k z k is suitably normalized and not too large for | z | = 1 , then its roots are clustered around | z | = 1 and equidistribute in angle at scale ∼ n − 1 / 2 . We establish a connection between the rate of equidistribution of roots in angle and the number of sign changes of the corresponding trigonometric polynomial q ( θ ) = ℜ ∑ k = 0 n a k e i k θ . If q ( θ ) has ≲ n δ roots for some 0 < δ < 1 / 2 , then the roots of p ( z ) do not frequently cluster in angle at scale ∼ n − ( 1 − δ ) ≪ n − 1 / 2 .