Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-29T11:44:39.926Z Has data issue: false hasContentIssue false

A NOTE ON A RESULT OF RUZSA, II

Published online by Cambridge University Press:  18 June 2010

MIN TANG*
Affiliation:
Department of Mathematics, Anhui Normal University, Wuhu 241000, PR China (email: tmzzz2000@163.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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].

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Chen, Y. G., ‘The analogue of Erdős–Turán conjecture in ℤm’, J. Number Theory 128 (2008), 25732581.CrossRefGoogle Scholar
[2]Erdős, P., ‘On a problem of Sidon in additive number theory’, Acta Sci. Math. (Szeged) 15 (1954), 255259.Google Scholar
[3]Erdős, P. and Turán, P., ‘On a problem of Sidon in additive number theory, and on some related problems’, J. Lond. Math. Soc. 16 (1941), 212215.CrossRefGoogle Scholar
[4]Panaitopol, L., ‘Inequalities concerning the function π(x): applications’, Acta Arith. 94(4) (2000), 373381.CrossRefGoogle Scholar
[5]Ruzsa, I. Z., ‘A just basis’, Monatsh. Math. 109 (1990), 145151.CrossRefGoogle Scholar
[6]Tang, M., ‘A note on a result of Ruzsa’, Bull. Aust. Math. Soc. 77 (2008), 9198.CrossRefGoogle Scholar