On the order of additive bases in finite abelian groups

Lê, Thái Hoàng; Qiu, Christopher

doi:10.2140/involve.2025.18.737

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On the order of additive bases in finite abelian groups

Thái Hoàng Lê and Christopher Qiu

Vol. 18 (2025), No. 4, 737–746

DOI: 10.2140/involve.2025.18.737

Abstract

Let $G$ be a finite abelian group. For a subset $A \subset G$ , we define ${ord}_{G}^{*} (A)$ to be the smallest number $h$ such that $G = h A$ and ${ord}_{G} (A)$ to be the smallest number $ℓ$ such that $G = ⋃_{i = 0}^{ℓ} i A$ , if these numbers exist. Here $h A$ is the $h$ -fold sumset of $A$ . We address and obtain some partial results on the following questions:

How large can ${ord}_{G}^{*} (A)$ be, in terms of $G$ ?
How small must ${ord}_{G}^{*} (A)$ be, in terms of ${ord}_{G} (A)$ and $G$ ?

These questions form part of a long line of research initiated by Erdős and Graham, who first studied the problems in $ℕ$ .

Keywords

additive basis, order of basis, Davenport constant, finite abelian group

Mathematical Subject Classification

Primary: 11B13, 20K01

Milestones

Received: 31 December 2023

Accepted: 5 April 2024

Published: 30 July 2025

Communicated by Vadim Ponomarenko

Authors

Thái Hoàng Lê
Department of Mathematics University of Mississippi University, MS United States

Christopher Qiu
Bridgewater-Raritan High School Bridgewater, NJ United States

Vol. 18, No. 4, 2025