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A NOTE ON A RESULT OF RUZSA

Part of: Sequences and sets

Published online by Cambridge University Press:  01 February 2008

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

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Let σA(n)=∣{(a,a′)∈A2:a+a′=n}∣, where and A is a subset of . Erdös and Turán conjectured that, for any basis A of , σA(n) is unbounded. In 1990, Ruzsa constructed a basis 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 satisfying for large enough N.

Keywords

Erdös–Turán conjecturebasis

MSC classification

Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 1 , February 2008 , pp. 91 - 98
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Erdös, P., ‘On a problem of Sidon in additive number theory’, Acta Sci. Math. (Szeged) 15 (1954), 255–259.Google Scholar
[2]Erdös, P. and Turán, P., ‘On a problem of Sidon in additive number theory, and on some related problems’, J. London Math. Soc. 16 (1941), 212–215.CrossRefGoogle Scholar
[3]Grekos, G., Haddad, L., Helou, C. and Pihko, J., ‘On the Erdös–Turán Conjecture’, J. Number Theory 102 (2003), 339–352.Google Scholar
[4]Nes˘etr˘il, J. and Serra, O., ‘The Erdös–Turán property for a class of bases’, Acta Arith. 115 (2004), 245–254.Google Scholar
[5]Borwein, P., Choi, S. and Chu, F., ‘An old conjecture of Erdös–Turán on additive bases’, Math. Comp. 75 (2005), 475–484.Google Scholar
[6]Ruzsa, I. Z., ‘A just basis’, Monatsh. Math. 109 (1990), 145–151.Google Scholar
[7]Tang, M. and Chen, Y. G., ‘A basis of ’, Colloq. Math. 104 (2006), 99–103.CrossRefGoogle Scholar
[8]Tang, M. and Chen, Y. G., ‘A basis of . II’, Colloq. Math. 108 (2007), 141–145.CrossRefGoogle Scholar