On a theorem of Erdos and Sarkozy

Let A = {a(1), a(2),...}(a(1) <= a(2) <= ...) be an infinite sequence of nonnegative integers, k >= 2 be a fixed integer and denote by R-k (n) the number of solutions of a(i1) + a(i2) + ... + a(ik) = n. In this paper, we prove that if g(n) is a monotonically increasing arithmetic function with g(n) -> +infinity and g(n) = o(n(log n)(-2)), then for any 0 < epsilon < 1, vertical bar R-k(n) - g(n)vertical bar > ([k/2]! - epsilon)root g(n) holds for infinitely many positive integers n. We also prove that for a positive integer d, if R-k(n) >= d for all sufficiently large integers n, then R-k(n) >= d + 2[k/2]!root d + ([k/2]!)(2) for infinitely many positive integers n.