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ON A PROBLEM OF CHEN AND LEV

  • SHI-QIANG CHEN (a1), MIN TANG (a2) and QUAN-HUI YANG (a3)
Abstract

For a given set $S\subset \mathbb{N}$ , $R_{S}(n)$ is the number of solutions of the equation $n=s+s^{\prime },s<s^{\prime },s,s^{\prime }\in S$ . Suppose that $m$ and $r$ are integers with $m>r\geq 0$ and that $A$ and $B$ are sets with $A\cup B=\mathbb{N}$ and $A\cap B=\{r+mk:k\in \mathbb{N}\}$ . We prove that if $R_{A}(n)=R_{B}(n)$ for all positive integers $n$ , then there exists an integer $l\geq 1$ such that $r=2^{2l}-1$ and $m=2^{2l+1}-1$ . This solves a problem of Chen and Lev [‘Integer sets with identical representation functions’, Integers16 (2016), A36] under the condition $m>r$ .

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The second author was supported by the National Natural Science Foundation of China, Grant No. 11471017. The third author was supported by the National Natural Science Foundation for Youth of China, Grant No. 11501299, the Natural Science Foundation of Jiangsu Province, Grant Nos. BK20150889 and 15KJB110014, and the Startup Foundation for Introducing Talent of NUIST, Grant No. 2014r029.

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[1] Chen, Y. G. and Lev, V. F., ‘Integer sets with identical representation functions’, Integers 16 (2016), A36, 4 pages.
[2] Dombi, G., ‘Additive properties of certain sets’, Acta Arith. 103 (2002), 137146.
[3] Erdős, P. and Sárközy, A., ‘Problems and results on additive properties of general sequences, I’, Pacific J. Math. 118 (1985), 347357.
[4] Erdős, P. and Sárközy, A., ‘Problems and results on additive properties of general sequences, II’, Acta Math. Hungar. 48 (1986), 201211.
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[8] Kiss, S. Z. and Sándor, C., ‘Partitions of the set of nonnegative integers with the same representation functions’, Discrete Math. 340 (2017), 11541161.
[9] Lev, V. F., ‘Reconstructing integer sets from their representation functions’, Electron. J. Combin. 11 (2004), R78, 6 pages.
[10] Li, J. W. and Tang, M., ‘Partitions of the set of nonnegative integers with the same representation functions’, Bull. Aust. Math. Soc. 97 (2018), 200206.
[11] Rozgonyi, E. and Sándor, C., ‘An extension of Nathanson’s theorem on representation functions’, Combinatorica 37 (2016), 117.
[12] Sándor, C., ‘Partitions of natural numbers and their representation functions’, Integers 4 (2004), A18, 5 pages.
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[14] Tang, M., ‘Partitions of natural numbers and their representation functions’, Chinese Ann. Math. Ser A 37 (2016), 4146; English translation, Chinese J. Contemp. Math. 37 (2016), 39–44.
[15] Yu, W. and Tang, M., ‘A note on partitions of natural numbers and their representation functions’, Integers 12 (2012), A53, 5 pages.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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