Théorie des nombres
On the structure of the $h$-fold sumsets
Comptes Rendus. Mathématique, Tome 359 (2021) no. 4, pp. 493-500.

Let $A$ be a set of nonnegative integers. Let ${\left(hA\right)}^{\left(t\right)}$ 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 $k\ge 2$, and $A=\left\{{a}_{0},{a}_{1},\cdots ,{a}_{k}\right\}$ is a finite set of integers such that $0={a}_{0}<{a}_{1}<\cdots <{a}_{k}$ and $gcd\left({a}_{1},{a}_{2},\cdots ,{a}_{k}\right)=1,$ then there exist integers ${c}_{t},{d}_{t}$ and sets ${C}_{t}\subseteq \left[0,{c}_{t}-2\right]$, ${D}_{t}\subseteq \left[0,{d}_{t}-2\right]$ such that

 ${\left(hA\right)}^{\left(t\right)}={C}_{t}\cup \left[{c}_{t},h{a}_{k}-{d}_{t}\right]\cup \left(h{a}_{k-1}-{D}_{t}\right)$

for all $h\ge {\sum }_{i=2}^{k}\left(t{a}_{i}-1\right)-1.$ This improves a recent result of Nathanson with the bound $h\ge \left(k-1\right)\left(t{a}_{k}-1\right){a}_{k}+1$.

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DOI : https://doi.org/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
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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},
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publisher = {Acad\'emie des sciences, Paris},
volume = {359},
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doi = {10.5802/crmath.191},
language = {en},
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Jun-Yu Zhou; Quan-Hui Yang. On the structure of the $h$-fold sumsets. Comptes Rendus. Mathématique, Tome 359 (2021) no. 4, pp. 493-500. doi : 10.5802/crmath.191. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.191/

[1] Andrew Granville; George Shakan The Frobenius postage stamp problem, and beyond (2020) (https://arxiv.org/abs/2003.04076) | Article | Zbl 07301143

[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 | Article | MR 304305

[4] Melvyn B. Nathanson Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, 165, Springer, 1996 | Zbl 0859.11003

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

[6] Jiandong Wu; Fengjuan Chen; Yonggao Chen On the structure of the sumsets, Discrete Math., Volume 311 (2011) no. 6, pp. 408-412 | MR 2799890 | Zbl 1225.11016

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