On a generalization of a theorem of Erdos and Fuchs

Let A = {a(1), a(2), ...} (a(1) < a(2) < ...) be an infinite sequence of nonnegative integers, let k >= 2 be a fixed integer and denote by r(k)(A, n) the number of solutions of a(i1) + a(i2) + ... + a(ik) <= n. Montgomery and Vaughan proved that r(2)(A, n) = cn+o(n(1/4)) cannot hold for any constant c > 0. In this paper, we extend this result to k > 2. (C) 2009 Elsevier B.V. All rights reserved.