On additive irreducibility of multiplicative subgroups

2025/04/14 by Kalmynin, Alexander · 3 citations
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.10202

Abstract

In this paper, we employ a version of Stepanov's method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order p. Specifically, we show that if a subgroup μ_d of d-th roots of unity satisfies A-A=μ_d∪\0\, then d=2 or 6. Additionally, we resolve the Sárközy's conjecture on quadratic residues: for prime p, the set \mathcal R_p of quadratic residues modulo p cannot be represented as A+B for A,B with min(|A|,|B|)>1. More generally, we prove that if the set of d-th roots of unity μ_d is represented non-trivially as A+B, then the sizes of summands are equal.

Cited by

• Multiplicative irreducibility of shifted multiplicative subgroups in the extremal case
• Multiplicative Subgroups of Prime Fields Are Not Sumsets
• Multiplicative subgroups are not restricted sumsets

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• Multiplicative irreducibility of shifted multiplicative subgroups in the extremal case
• Multiplicative Subgroups of Prime Fields Are Not Sumsets
• Multiplicative subgroups are not restricted sumsets
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• On some polynomial version on the sum-product problem for subgroups