Sumsets and generalized arithmetic progressions in multiplicative subgroups
Number Theory
2026-07-30 v1 Combinatorics
Abstract
Let , and let be a multiplicative subgroup with . We prove that a proper subgroup is a generalized arithmetic progression (GAP) if and only if , and we determine when the full group is a GAP. For certain families of subgroups, we obtain the stronger conclusion that is additively irreducible. In particular, if and for some , then admits no nontrivial sumset decomposition. We also prove that every has fewer than representations as a sum (or difference) of two elements of whenever and , which may be of independent interest.
Cite
@article{arxiv.2607.28559,
title = {Sumsets and generalized arithmetic progressions in multiplicative subgroups},
author = {Albert Cochrane},
journal= {arXiv preprint arXiv:2607.28559},
year = {2026}
}
Comments
16 pages