A generalization of the Erdös–Surányi problem.

Erdös–Surányi and Prielipp suggested to study the following problem: For any integers k > 0 and n , are there an integer N and a map ϵ : { 1 , … , N } → { − 1 , 1 } such that (0.1) n = ∑ j = 1 N ϵ ( j ) j k ? Mitek and Bleicher independently solved this problem affirmatively. In this paper we consider the case that for some positive odd integer L the numbers ϵ ( j ) are L -th roots of unity. We show that the answer to the corresponding question is negative if and only if L is a prime power. [ABSTRACT FROM AUTHOR]