A NOTE ON A RESULT OF RUZSA

Let σ A ( n )=∣{( a , a′ )∈ A 2 : a + a′ = n }∣, where n\in \mathbb {N}and A is a subset of \mathbb {N}. Erdös and Turán conjectured that, for any basis A of \mathbb {N}, σ A ( n ) is unbounded. In 1990, Ruzsa constructed a basis A\subset \mathbb {N}for which σ A ( n ) is bounded in the square mean. In this paper, based on Ruzsa’s method, we show that there exists a basis A of \mathbb {N}satisfying $\sum _{n\leq N}\sigma...