Let be a set of nonnegative integers. Let be the set of all integers in the sumset that have at least representations as a sum of elements of . In this paper, we prove that, if , and is a finite set of integers such that and then there exist integers and sets , such that
for all This improves a recent result of Nathanson with the bound .
Revised:
Accepted:
Published online:
@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://archive.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://archive.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 %I Académie des sciences, Paris %U http://archive.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://archive.numdam.org/articles/10.5802/crmath.191/
[1] The Frobenius postage stamp problem, and beyond (2020) (https://arxiv.org/abs/2003.04076) | DOI | Zbl
[2] A tight structure theorem for sumsets (2020) (https://arxiv.org/abs/2006.01041)
[3] Sums of finite sets of integers, Am. Math. Mon., Volume 79 (1972), pp. 1010-1012 | DOI | MR
[4] Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Texts in Mathematics, 165, Springer, 1996 | Zbl
[5] Sums of finite sets of integers, II (2020) (https://arxiv.org/abs/2005.10809v3)
[6] On the structure of the sumsets, Discrete Math., Volume 311 (2011) no. 6, pp. 408-412 | MR | Zbl
Cited by Sources: