Generalized additive bases and difference bases for Cartesian product of finite abelian groups

2025/09/28 by Li, Shuxing, Yip, Chi Hoi
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.24034

Abstract

For a finite group G and positive integer g, a g-additive basis is a subset of G whose pairwise sums cover each element of G at least g times, with g-difference bases defined similarly using pairwise differences. While prior work focused on 1-additive and 1-difference bases, recent works of Kravitz and Schmutz--Tait explored g-additive and g-difference bases in finite abelian groups. This paper investigates such bases in Gⁿ, the Cartesian product of a finite abelian group G. We construct g-additive and g-difference bases in Gⁿ, which lead to asymptotically sharp upper bounds on the minimal sizes of such bases. Our proofs draw on ideas from additive combinatorics and combinatorial design theory.

Citations

• Cardinalities of g-difference sets
• Finite field models in arithmetic combinatorics -- twenty years on
• Formal Duality in Finite Abelian Groups
• Semifields, relative difference sets, and bent functions
• Generalized Sidon sets
• Finite field models in additive combinatorics
• A Complete Annotated Bibliography of Work Related to Sidon Sequences
• Bases and decomposition numbers of finite groups

Related

• On the additive bases problem in finite fields
• Additive bases in groups
• Additive Bases in Abelian Groups
• A Walk Through Some Newer Parts of Additive Combinatorics
• A connection between matchings and removal in abelian groups