On additive irreducibility of multiplicative subgroups

TL;DR

This work investigates the tension between multiplicative structure and additive structure in finite fields by leveraging Hanson–Petridis Stepanov-type polynomials. It proves that additively reducible multiplicative subgroups $\mu_d$ can occur only for the small cases $d\in\{2,6\}$, resolves the Lev–Sonn conjecture, and establishes the $\alpha=\beta$ structure constraint for representations $\mu_d=A+B$. It then proves Sárközy’s conjecture that the quadratic residues are not additively reducible, unifying several conjectures in additive combinatorics for prime fields. The results advance understanding of additive structure within multiplicative subgroups and demonstrate the power of HP polynomials paired with differential-form techniques to derive strong, general irreducibility statements.

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 $A-A=μ_d\cup\{0\}$, 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.

Theorems & Definitions (51)

• Theorem 1: Lev-Sonn conjecture
• Proposition 1
• proof
• Theorem 2: $\alpha=\beta$ theorem
• Corollary 1
• proof
• Theorem 3
• Definition 1: Hanson-Petridis polynomial
• Lemma 1: Explicit formula for coefficients
• proof