A NOTE ON A RESULT OF RUZSA

Abstract

AbstractLet σA(n)=∣{(a,a′)∈A2: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 $\mathbb {N}$, σA(n) is unbounded. In 1990, Ruzsa constructed a basis $A\subset \mathbb {N}$ for which σA(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.