Browse > Article
http://dx.doi.org/10.4134/BKMS.b200054

SOME INVERSE RESULTS OF SUMSETS  

Tang, Min (School of Mathematics and Statistics Anhui Normal University)
Xing, Yun (School of Mathematics and Statistics Anhui Normal University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.2, 2021 , pp. 305-313 More about this Journal
Abstract
Let h ≥ 2 and A = {a0, a1, …, ak-1} be a finite set of integers. It is well-known that |hA| = hk - h + 1 if and only if A is a k-term arithmetic progression. In this paper, we give some nontrivial inverse results of the sets A with some extremal the cardinalities of hA.
Keywords
Sumsets; inverse problem; arithmetic progression;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Y.-G. Chen, On addition of two sets of integers, Acta Arith. 80 (1997), no. 1, 83-87. https://doi.org/10.4064/aa-80-1-83-87   DOI
2 G. A. Freiman, The addition of finite sets. I, Izv. Vyss. Ucebn. Zaved. Matematika 1959, no. 6 (13), 202-213.
3 G. A. Freiman, Inverse problems of additive number theory. VI. On the addition of finite sets. III, Izv. VysS. Ucebn. Zaved. Matematika 1962, no. 3 (28), 151-157.
4 G. A. Freiman, On the detailed structure of sets with small additive property, in Combinatorial number theory and additive group theory, 233-239, Adv. Courses Math. CRM Barcelona, Birkhauser Verlag, Basel, 2009. https://doi.org/10.1007/978-3-7643-8962-8_17   DOI
5 G. A. Freiman, Inverse additive number theory. XI. Long arithmetic progressions in sets with small sumsets, Acta Arith. 137 (2009), no. 4, 325-331. https://doi.org/10.4064/aa137-4-2   DOI
6 V. F. Lev, Restricted set addition in groups. I. The classical setting, J. London Math. Soc. (2) 62 (2000), no. 1, 27-40. https://doi.org/10.1112/S0024610700008863   DOI
7 V. F. Lev, Structure theorem for multiple addition and the Frobenius problem, J. Number Theory 58 (1996), no. 1, 79-88. https://doi.org/10.1006/jnth.1996.0065   DOI
8 V. F. Lev and P. Y. Smeliansky, On addition of two distinct sets of integers, Acta Arith. 70 (1995), no. 1, 85-91. https://doi.org/10.4064/aa-70-1-85-91   DOI
9 M. B. Nathanson, Sums of finite sets of integers, Amer. Math. Monthly 79 (1972), 1010-1012. https://doi.org/10.2307/2318072   DOI
10 M. B. Nathanson, Inverse theorems for subset sums, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1409-1418. https://doi.org/10.2307/2154821   DOI
11 M. B. Nathanson, Additive Number Theory, Graduate Texts in Mathematics, 165, Springer-Verlag, New York, 1996. https://doi.org/10.1007/978-1-4757-3845-2   DOI
12 Y. Stanchescu, On addition of two distinct sets of integers, Acta Arith. 75 (1996), no. 2, 191-194. https://doi.org/10.4064/aa-75-2-191-194   DOI
13 Q.-H. Yang and Y.-G. Chen, On the cardinality of general h-fold sumsets, European J. Combin. 47 (2015), 103-114. https://doi.org/10.1016/j.ejc.2015.02.002   DOI