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A NOTE ON A RESULT OF RUZSA, II

Part of: Sequences and sets

Published online by Cambridge University Press:  18 June 2010

MIN TANG
MIN TANG*
Affiliation:
Department of Mathematics, Anhui Normal University, Wuhu 241000, PR China (email: [email protected])
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Abstract

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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 ∑ n≤Nσ2A(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 ∑ n≤Nσ2A(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].

Keywords

Erdős–Turán conjecturebasis

MSC classification

Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 340 - 347
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

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