A just basis

Abstract

An old problem of P. Erdös and P. Turán asks whether there is a basisA of order 2 for which the number of representationsn=a+a′, a,a′∈A is bounded. Erdős conjectured that such a basis does not exist. We answer a related finite problem and find a basis for which the number of representations is bounded in the square mean. Writing σ (n)=|{(a, a ^t) ∈A ²:a+a′=n}| we prove that there exists a setA of nonnegative integers that forms a basis of order 2 (that is,s(n)≥1 for alln), and satisfies ∑_{n ⩽ N} σ(N)² = O(N).