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 $G$, 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.

22pages, title updated. This version contains substantially stronger results