×

A new upper bound on Ruzsa’s numbers on the Erdős-Turán conjecture. (English) Zbl 07872247

Summary: In this paper, we show that the Ruzsa number \(R_{m}\) is bounded by 192 for any positive integer \(m\), which improves the prior bound \(R_{m} \leq 288\) given by Chen in 2008.

MSC:

11B13 Additive bases, including sumsets
11B34 Representation functions
Full Text: DOI

References:

[1] Borwein, P., Choi, S. and Chu, F., An old conjecture of Erdős-Turán on additive bases, Math. Comp.75 (2005) 475-484. · Zbl 1093.11018
[2] Chen, Y.-G., The analogue of Erdős-Turán conjecture in \(\Bbb Z_m\), J. Number Theory128 (2008) 2573-2581. · Zbl 1191.11006
[3] Chen, Y.-G., On the Erdős-Turán conjecture, C. R. Math. Acad. Sci. Paris350 (2012) 933-935. · Zbl 1279.11012
[4] Chen, Y.-G. and Sun, T., The difference basis and bi-basis of \(\Bbb Z_m\), J. Number Theory130 (2010) 716-726. · Zbl 1217.11012
[5] 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) 212-215. · Zbl 0061.07301
[6] Halberstam, H. and Roth, K. F., Sequences (Clarendon Press, Oxford, 1966). · Zbl 0141.04405
[7] Panaitopol, L., Inequalities concerning the function \(\pi(x)\): Applications, Acta Arith.94 (2000) 373-381. · Zbl 0963.11050
[8] Ruzsa, I. Z., A just basis, Monatsh. Math.109 (1990) 145-151. · Zbl 0713.11009
[9] Grekos, G., Haddad, L., Helou, C. and Pihko, J., On the Erdős-Turán conjecture, J. Number Theory102 (2003) 339-352. · Zbl 1083.11010
[10] Sándor, C., A note on a conjecture of Erdős-Turán, Integers8 (2008) A30. · Zbl 1210.11017
[11] Sándor, C. and Yang, Q.-H., A lower bound of Ruzsa’s number related to the Erdős-Turán conjecture, Acta Arith.180 (2017) 161-169. · Zbl 1425.11023
[12] Tang, M. and Chen, Y.-G., A basis of \(\Bbb Z_m\), Colloq. Math.104 (2006) 99-103. · Zbl 1138.11003
[13] Tang, M. and Chen, Y.-G., A basis of \(\Bbb Z_m\), II, Colloq. Math.108 (2007) 141-145. · Zbl 1187.11006
[14] Tao, T. and Van Vu, H., Additive Combinatorics, , Vol. 105 (Cambridge University Press, 2010). · Zbl 1179.11002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.