Number theory
On the structure of the $h$-fold sumsets
Comptes Rendus. Mathématique, Volume 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 $h\ge \left(k-1\right)\left(t{a}_{k}-1\right){a}_{k}+1$.

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
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title = {On the structure of the $h$-fold sumsets},
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.191/

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[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

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