PreprintCC BY 4.0arXiv
Gaps in the Most Frequent Sizes of Integer Sumsets
Abstract
Nathanson observed that, for four-element sets A of integers, the most frequent sizes of the iterated sumset hA are separated by triangular numbers. Senger proved this and suggested an analogue for larger sets. We establish this pattern for k-element subsets of {1,…,q}, for every fixed k≥3, where h≥2 is either fixed or grows slower than q1/(2k−1). As q→∞, almost every such set has the largest possible h-fold sumset, and almost every exception has one of h−1 explicit smaller sizes. The differences between consecutive sizes among these h values are consecutive (k−2)-dimensional simplex numbers.
Related
BibTeX
@misc{awan2026gaps,
title = {{Gaps in the Most Frequent Sizes of Integer Sumsets}},
author = {Saad A. Awan and Steven J. Miller and Linh Nguyen and Steven Senger and Elvis Wong},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07303},
eprint = {2611.07303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07303},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaps in the Most Frequent Sizes of Integer Sumsets
AU - Saad A. Awan
AU - Steven J. Miller
AU - Linh Nguyen
AU - Steven Senger
AU - Elvis Wong
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07303
UR - https://arxiv.org/abs/2611.07303
AB - Nathanson observed that, for four-element sets A of integers, the most frequent sizes of the iterated sumset hA are separated by triangular numbers. Senger proved this and suggested an analogue for larger sets. We establish this pattern for k-element subsets of {1,…,q}, for every fixed k≥3, where h≥2 is either fixed or grows slower than q^1/(2k−1). As q→∞, almost every such set has the largest possible h-fold sumset, and almost every exception has one of h−1 explicit smaller sizes. The differences between consecutive sizes among these h values are consecutive (k−2)-dimensional simplex numbers.
ER -