Ruzsa’s theorem on Erdős and Turán conjecture

For any set A of nonnegative integers, let σ A ( n ) be the number of solutions to the equation n = a + b , a , b ∈ A . The set A is called a basis of N if σ A ( n ) ≥ 1 for all n ≥ 1 . The well known Erdős–Turán conjecture says that if A is a basis of N , then σ A ( n ) cannot be bounded. In 1990, Ruzsa proved that there exists a basis A of N such that ∑ n ≤ N σ A 2 ( n ) = O ( N ) . In this paper, we give a new proof of Ruzsa’s Theorem.