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arXiv · 2602.20919

Multiplicative irreducibility of shifted multiplicative subgroups

Seoyoung Kim·Chi Hoi Yip·Semin Yoo
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Abstract

In a recent breakthrough, Kalmynin resolved conjectures of Lev--Sonn and Sárközy on additive decompositions of multiplicative subgroups of prime fields. In this paper, inspired by a related conjecture of Sárközy, we prove multiplicative analogues of Kalmynin's results. We show that for every proper multiplicative subgroup $G$, the shifted set $(G-1)\setminus\{0\}$ cannot be written as a product set nontrivially, addressing a conjecture of Sárközy. In addition, we prove that no nonzero shift of any coset of a proper multiplicative subgroup is a ratio set of the form $A/A$. Our results substantially sharpen previous theorems of Shkredov and the authors.

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Seoyoung Kim, Chi Hoi Yip, Semin Yoo. 2026-03-13. Multiplicative irreducibility of shifted multiplicative subgroups. https://doi.org/10.1112/jlms.70688

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