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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
Abstract

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

  1. How large can ord G(A) be, in terms of G?

  2. 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 .

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