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arXiv · 2509.24034

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

Shuxing Li·Chi Hoi Yip
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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 $G^n$, the Cartesian product of a finite abelian group $G$. We construct $g$-additive and $g$-difference bases in $G^n$, 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.

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Shuxing Li, Chi Hoi Yip. 2025-09-28. Generalized additive bases and difference bases for Cartesian product of finite abelian groups. https://arxiv.org/abs/2509.24034

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