Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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 builds upon recent work studying additive decompositions of subgroups, and we generalize a result of Hanson and Petridis to show that any additive $n$-decomposition of a subgroup must be a direct sum. We also show how this classification quickly follows from Kalmynin’s recent work resolving Sárközy’s conjecture for quadratic residues.