Multiplicative irreducibility of shifted multiplicative subgroups

TL;DR

This paper proves that shifted multiplicative subgroups in prime fields cannot be decomposed into product or ratio sets, advancing understanding of their algebraic structure and confirming conjectures related to their irreducibility.

Contribution

It establishes multiplicative analogues of Kalmynin's results, showing shifted subgroups are multiplicatively irreducible, and refines previous theorems on their algebraic properties.

Findings

Shifted subgroups cannot be expressed as nontrivial product sets.

No nonzero shift of a coset is a ratio set of the form A/A.

Results sharpen previous theorems of Shkredov and the authors.

Abstract

In a recent breakthrough, Kalmynin resolved conjectures of Lev--Sonn and S\'{a}rk\"{o}zy on additive decompositions of multiplicative subgroups of prime fields. In this paper, inspired by a related conjecture of S\'{a}rk\"{o}zy, 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 S\'{a}rk\"{o}zy. 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.