Ruzsa's theorem on Erdos and Turan conjecture

For any set A of nonnegative integers, let sigma(A) (n) be the number of solutions to the equation n = a+b, a, b is an element of A. The set A is called a basis of N if sigma(A)(n) >= 1 for all n >= 1. The well known Erdos-Turan conjecture says that if A is a basis of N. then sigma(A)(n) cannot be bounded. In 1990, Ruzsa proved that there exists a basis A of N such that Sigma(n <= N)sigma(2)(A) (n) = O(N). In this paper, we give a new proof of Ruzsa's Theorem. (c) 2012 Elsevier Ltd. All rights reserved.