On sums of two and three roots of unity

Abstract

Let α≠0 be a sum of some k distinct Nth roots of unity, where 2≤k<N. In 1986, Myerson raised the following two problems. How small can |α| be? How large can the modulus of the product of all conjugates of α lying in the disc |z|<1 be? A simple Liouville type argument gives the lower bound k−N+2 for these quantities, so the problem is to find appropriate upper bounds. As for the first question, for k≥5, it remains a huge gap between lower and the best known upper bound N−dk. In this note, we give a complete answer to the second question of Myerson for k=2. For k=3 and N large prime, we show that a positive proportion of the conjugates of any such α lie in the disc |z|≤ϱ, where ϱ<1. This implies a naturally expected upper bound.