English

Sumsets and generalized arithmetic progressions in multiplicative subgroups

Number Theory 2026-07-30 v1 Combinatorics

Abstract

Let q=pfq=p^f, and let AFq×A\leq\mathbb{F}_q^\times be a multiplicative subgroup with Fp(A)=Fq\mathbb{F}_p(A)=\mathbb{F}_q. We prove that a proper subgroup AA is a generalized arithmetic progression (GAP) if and only if A{1,2,4}|A| \in \{1, 2, 4\}, and we determine when the full group Fq×\mathbb{F}_q^\times is a GAP. For certain families of subgroups, we obtain the stronger conclusion that AA is additively irreducible. In particular, if A>4|A|>4 and pe1(modA)p^e\equiv-1\pmod{|A|} for some e1e\ge1, then AA admits no nontrivial sumset decomposition. We also prove that every c0c \neq 0 has fewer than A/2|A|/2 representations as a sum (or difference) of two elements of AA whenever [Fq×:A]3[\mathbb{F}_q^\times:A] \ge3 and A5|A| \ge 5, 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}
}

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16 pages