Multiplicative Subgroups of $\mathbb{Z}_p^*$ that are Generalized Arithmetic Progressions

TL;DR

We tackle the question of when a multiplicative subgroup $A_k=\{x^k:x\in\mathbb Z_p^*\}$ of the finite field $\mathbb Z_p$ can carry the additive structure of a generalized arithmetic progression (GAP). The authors develop an additive-decomposition framework, show that any GAP structure forces a direct-sum decomposition into $2$-element components, and then rule out almost all cases by combining 2-decomposition results, doubling arguments, and bounds from projective Fermat curves via the Hasse-Weil estimates. Consequently, they prove that $A_k$ is a GAP if and only if $|A_k|\in\{2,4,p-1\}$. This dichotomy highlights the rigidity of sum-product phenomena in finite fields and precisely characterizes when multiplicative subgroups can admit additive generalized-structure.

Abstract

We prove that a multiplicative subgroup $A_k$ of $\mathbb{Z}_p^*$ is a generalized arithmetic progression if and only if $|A_k| = 2,\ 4,$ or $p-1$. Much of the argument is built upon recent work studying additive decompositions of subgroups of $\mathbb{Z}_p^*$, and we generalize a result of Hanson and Petridis to show that any additive $n$-decomposition of a subgroup must be a direct sum.

Theorems & Definitions (15)

• Theorem 1.1
• Lemma 4.1
• Lemma 4.2: ha; Corollary 1.3
• Theorem 4.1: gy; Theorem 1.2
• Corollary 4.1
• proof
• Corollary 4.2
• proof
• Lemma 5.1: che; Lemma 2.6
• Lemma 5.2