Sumsets and generalized arithmetic progressions in multiplicative subgroups

2026/07/30 by Albert Cochrane
Mathematics · #math.CO #math.NT #msc:11B13 #msc:11B25 #msc:11B30 #msc:11T06

paper · pdf

16 pages

Abstract

Let q=p^f, and let A≤\mathbbF_q^× be a multiplicative subgroup with \mathbbF_p(A)=\mathbbF_q. We prove that a proper subgroup A is a generalized arithmetic progression (GAP) if and only if |A| ∈ \1, 2, 4\, and we determine when the full group \mathbbF_q^× is a GAP. For certain families of subgroups, we obtain the stronger conclusion that A is additively irreducible. In particular, if |A|>4 and p^e≡-1\pmod|A| for some e≥1, then A admits no nontrivial sumset decomposition. We also prove that every c ≠ 0 has fewer than |A|/2 representations as a sum (or difference) of two elements of A whenever [\mathbbF_q^×:A] ≥3 and |A| ≥ 5, which may be of independent interest.

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• Multiplicative Subgroups of Prime Fields Are Not Sumsets
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