Abstract
Let \(\mathbb {N}\) be the set of nonnegative integers. For a given set \(S\subset \mathbb {N}\) the representation function \(R_S(n)\) is defined as the number of solutions of the equation \(n=s+s', s<s', s,s'\in S\). In this paper, we characterize some sets which have the same representation functions.
Similar content being viewed by others
References
G. Dombi, Additive properties of certain sets. Acta Arith. 103, 137–146 (2002)
V.F. Lev, Reconstructing integer sets from their representation functions. Electron. J. Combin. 11, R78 (2004)
C. Sándor, Partitions of natural numbers and their representation functions. Integers 4, A18 (2004)
M. Tang, Partitions of the set of natural numbers and their representation functions. Discrete Math. 308, 2614–2616 (2008)
W. Yu, M. Tang, A note on partitions of natural numbers and their representation functions. Integers 12, A53 (2012)
M. Tang, Partitions of natural numbers and their representation functions. Chinese Ann. Math. Ser. A 37, (2016), 41–46. For English version, see. Chin. J. Contemp. Math. 37, 39–44 (2016)
Y.G. Chen, V.F. Lev, Integer sets with identical representation functions. Integers 16, A36 (2016)
S.Z. Kiss, C. Sándor, Partitions of the set of nonnegative integers with the same representation functions. Discrete Math. 340, 1154–1161 (2017)
Y.G. Chen, On the values of representation functions. Sci. China Math. 54, 1317–1331 (2011)
Y.G. Chen, M. Tang, Partitions of natural numbers with the same representation functions. J. Number Theory 129, 2689–2695 (2009)
M.B. Nathanson, Representation functions of sequences in additive number theory. Proc. Am. Math. Soc. 72, 16–20 (1978)
Q.-H. Yang, Yong-Gao Chen, Partitions of natural numbers with the same weighted representation functions. J. Number Theory 132, 3047–3055 (2012)
S.Z. Kiss, C. Sándor, On the structure of sets which has coinciding representation functions, arXiv:1702.04499v1
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by National Natural Science Foundation of China, Grant No. 11471017.
Rights and permissions
About this article
Cite this article
Tang, M., Li, JW. On the structure of some sets which have the same representation functions. Period Math Hung 77, 232–236 (2018). https://doi.org/10.1007/s10998-018-0240-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-018-0240-5