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Bulletin of the London Mathematical Society Advance Access originally published online on February 6, 2009
Bulletin of the London Mathematical Society 2009 41(2):274-280; doi:10.1112/blms/bdn123
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© 2009 London Mathematical Society

On a question of Sárkozy and Sós for bilinear forms

Javier Cilleruelo

Departamento de Matemáticas
Universidad Autónoma de Madrid
28049 Madrid
Spain

Juanjo Rué

Departament de Matemàtica Aplicada II
Universitat Politècnica de Catalunya
Jordi Girona 1–3
08034 Barcelona
Spain
juan.jose.rue@upc.edu

Received 18 May 2008. Revision received 15 October 2008.

We prove that, if 2 ≤ k1 ≤ k2, then there is no infinite sequence A of positive integers such that the representation function r(n) = #{(a, a'): n = k1a + k2a', a, a' isin A} is constant for n large enough. This result completes the previous work of Dirac and Moser for the special case k1 = 1 and answers a question posed by Sárkozy and Sós.

2000 Mathematics Subject Classification 11B34.


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