LAUSR

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 − d k . 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.