Generalized additive bases and difference bases for Cartesian product of finite abelian groups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912613110120448 |
|---|---|
| author | Li, Shuxing Yip, Chi Hoi |
| author_facet | Li, Shuxing Yip, Chi Hoi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24034 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized additive bases and difference bases for Cartesian product of finite abelian groups Li, Shuxing Yip, Chi Hoi Combinatorics Number Theory 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. |
| title | Generalized additive bases and difference bases for Cartesian product of finite abelian groups |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2509.24034 |