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 (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 k2, 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
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
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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/

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

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