Theoretical analysis demonstrates multiplicative irreducibility of shifted subgroups in prime fields, highlighting structural constraints on product sets.
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 , the shifted set 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 . Our results substantially sharpen previous theorems of Shkredov and the authors.
No takes yet. Share an insight, caveat, or question.
Kim et al. (2026) studied this question.
Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context: