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
Zhou, Jun-Yu 1; Yang, Quan-Hui 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
@article{CRMATH_2021__359_4_493_0,
author = {Zhou, Jun-Yu and Yang, Quan-Hui},
title = {On the structure of the $h$-fold sumsets},
journal = {Comptes Rendus. Math\'ematique},
pages = {493--500},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {4},
year = {2021},
doi = {10.5802/crmath.191},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.191/}
}
TY  - JOUR
AU  - Zhou, Jun-Yu
AU  - Yang, Quan-Hui
TI  - On the structure of the $h$-fold sumsets
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 493
EP  - 500
VL  - 359
IS  - 4
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.191/
DO  - 10.5802/crmath.191
LA  - en
ID  - CRMATH_2021__359_4_493_0
ER  - 
%0 Journal Article
%A Zhou, Jun-Yu
%A Yang, Quan-Hui
%T On the structure of the $h$-fold sumsets
%J Comptes Rendus. Mathématique
%D 2021
%P 493-500
%V 359
%N 4
%U http://www.numdam.org/articles/10.5802/crmath.191/
%R 10.5802/crmath.191
%G en
%F CRMATH_2021__359_4_493_0
Zhou, Jun-Yu; Yang, Quan-Hui. On the structure of the $h$-fold sumsets. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 493-500. doi : 10.5802/crmath.191. http://www.numdam.org/articles/10.5802/crmath.191/

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

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

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

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

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

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

Cited by Sources: