Multiplicative irreducibility of shifted multiplicative subgroups

Abstract: A central theme in additive combinatorics is the interplay between addition and multiplication. Roughly speaking, sets with strong multiplicative structure are not expected to exhibit rich additive structure, and vice versa. In a recent breakthrough, Kalmynin resolved a conjecture of Lev--Sonn and Sárközy on additive decompositions of multiplicative subgroups in prime fields and quadratic residues. Motivated by this work, we study multiplicative analogues of these questions. We show that, under a certain additional condition, a shifted multiplicative subgroup cannot be written as a product set, and that it also cannot be written as a ratio set unconditionally. In this talk, I will discuss these results and the main ideas of the proofs. This talk is based on joint work with Seoyoung Kim (University of Basel) and Chi Hoi Yip (Georgia Institute of Technology).

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)