Multiplicative irreducibility of shifted multiplicative subgroups

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 , 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.