A NOTE ON A RESULT OF RUZSA

Source abstract

Abstract 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 $\sum _{n\leq N}\sigma _A(n)^2\leq 1\,449\,757\,928N$ for large enough N .