Multiplicative irreducibility of shifted multiplicative subgroups
Abstract
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.
Fields of Study & Subjects
Limits and Structures in Graph Theory
Analytic Number Theory Research
Advanced Combinatorial Mathematics
Publication details
- Journal / source
- Journal of the London Mathematical Society
- Volume
- 114
- Issue
- 3
- Publication date
- 27 August 2026
- Language
- EN