2026/07/27 by Semin Yoo
Mathematics · #math.CO #math.NT
In a recent breakthrough, Kalmynin proved a conjecture of Sárközy on additive irreducibility of the set of quadratic residues in a prime field. More recently, Kim, Yip, and Yoo initiated the study of a multiplicative analogue of the conjecture for shifted multiplicative subgroups. Specifically, they showed that for an odd prime p, a proper multiplicative subgroup G of \mathbb Fp^*, and λ∈ G, there do not exist sets A,B⊆ \mathbb Fp^* with |A|,|B|≥ 2 such that AB=(G-λ)∖\0\. In this paper, when λ∈ \mathbb Fp^* ∖ G, we completely resolve this problem in the equality case from a Stepanov bound in a prime field.