Bulletin of the Australian Mathematical Society

Research Article

A NOTE ON A RESULT OF RUZSA, II

MIN TANGa1

a1 Department of Mathematics, Anhui Normal University, Wuhu 241000, PR China (email: tmzzz2000@163.com)

Abstract

Let σA(n)=∣{(a,a′)∈A2:a+a′=n}∣, where n∈ℕ and A is a subset of ℕ. Erdős and Turán con-jectured that for any basis A of ℕ, σA(n) is unbounded. In 1990, Ruzsa constructed a basis A⊂ℕ for which σA(n) is bounded in square mean. Based on Ruzsa’s method, we proved that there exists a basis A of ℕ satisfying ∑ nNσ2A(n)≤1449757928N for large enough N. In this paper, we give a quantitative result for the existence of N, that is, we show that there exists a basis A of ℕ satisfying ∑ nNσ2A(n)≤1069693154N for N≥7.628 517 798×1027, which improves earlier results of the author [‘A note on a result of Ruzsa’, Bull. Aust. Math. Soc. 77 (2008), 91–98].

(Received February 21 2010)

2000 Mathematics subject classification

  • primary 11B13

Keywords and phrases

  • Erdős–Turán conjecture;
  • basis

Footnotes

The author was supported by the National Natural Science Foundation of China, Grant No. 10901002.