Comptes Rendus
Number theory
On the structure of the h-fold sumsets
Jun-Yu Zhou1; Quan-Hui Yang2 
1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2 School of Mathematics and Statistics, Nanjing University of Information, Science and Technology, Nanjing 210044, China
Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 493-500

Let A be a set of nonnegative integers. Let (hA) (t) be the set of all integers in the sumset hA that have at least t representations as a sum of h elements of A. In this paper, we prove that, if k≥2, and A=a 0 ,a 1 ,⋯,a k is a finite set of integers such that 0=a 0 <a 1 <⋯<a k and gcda 1 ,a 2 ,⋯,a k =1, then there exist integers c t ,d t and sets C t ⊆[0,c t -2], D t ⊆[0,d t -2] such that

(hA) (t) =C t ∪c t ,ha k -d t ∪ha k-1 -D t

for all h≥∑ i=2 k (ta i -1)-1. This improves a recent result of Nathanson with the bound h≥(k-1)(ta k -1)a k +1.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.191
Classification: 11B13

Jun-Yu Zhou 1; Quan-Hui Yang 2

1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2 School of Mathematics and Statistics, Nanjing University of Information, Science and Technology, Nanjing 210044, China
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     author = {Jun-Yu Zhou and Quan-Hui Yang},
     title = {On the structure of the $h$-fold sumsets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {493--500},
     year = {2021},
     publisher = {Acad\'emie des sciences, Paris},
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     doi = {10.5802/crmath.191},
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Jun-Yu Zhou; Quan-Hui Yang. On the structure of the $h$-fold sumsets. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 493-500. doi: 10.5802/crmath.191

[1] Andrew Granville; George Shakan The Frobenius postage stamp problem, and beyond (2020) (https://arxiv.org/abs/2003.04076) | DOI | Zbl

[2] Andrew Granville; Aled Walker A tight structure theorem for sumsets (2020) (https://arxiv.org/abs/2006.01041)

[3] Melvyn B. Nathanson Sums of finite sets of integers, Am. Math. Mon., Volume 79 (1972), pp. 1010-1012 | DOI | MR | Zbl

[4] Melvyn B. Nathanson Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, 165, Springer, 1996 | Zbl

[5] Melvyn B. Nathanson Sums of finite sets of integers, II (2020) (https://arxiv.org/abs/2005.10809v3) | Zbl

[6] Jiandong Wu; Fengjuan Chen; Yonggao Chen On the structure of the sumsets, Discrete Math., Volume 311 (2011) no. 6, pp. 408-412 | MR | Zbl

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