Heights of squares of Littlewood polynomials and infinite series

Abstract

Let P be a unimodular polynomial of degree d−1. Then the height of its square H(P ) is at least √ d/2 and the product L(P )H(P ), where L denotes the length of a polynomial, is at least d. We show that for any e > 0 and any d > d(e) there exists a polynomial P with ±1 coefficients of degree d − 1 such that H(P ) 1 and that there exist some infinite series with ±1 coefficients and an integer m(e) such that |Am| m(e).