Generalized additive bases and difference bases for Cartesian product of finite abelian groups (2509.24034v1)

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.

• The paper presents a framework for defining g-additive and g-difference bases that represent group elements multiple times.
• It establishes asymptotic bounds on the minimal sizes of these bases in Cartesian products of finite abelian groups using combinatorial methods.
• The study extends classical constructions like Sidon sets, providing insights for future computational applications in additive combinatorics.

Generalized Additive Bases and Difference Bases for Cartesian Products of Finite Abelian Groups

Introduction

The paper explores the notion of $g$-additive and $g$-difference bases within Cartesian products of finite abelian groups. A $g$-additive basis is a subset of a group $G$ such that every element can be expressed as the sum of pairs from the subset at least $g$ times. Analogously, a $g$-difference basis consists of pairs whose differences cover each element of $G$ at least $g$ times. These concepts extend beyond the traditional $1$-additive and $1$-difference bases, offering new combinatorial insights. This research aims to determine limits on the minimal sizes of such bases as the Cartesian product increases, leveraging additive combinatorics and combinatorial design theory.

Main Concepts

The primary mathematical constructs addressed in this paper include:

• Sidon Sets and Difference Sets: Sidon sets are defined by the constraint $r_{A+A}(x) \leq 2$ for all $x \in G$. This ensures maximal distinctiveness of representations by two terms from the set $A$. Similarly, difference sets represent every nonidentity element of $G$ exactly $g$ times as a difference of elements from a set $B$.
• Generalizations to $G^n$:
  • The paper investigates $g$-additive and $g$-difference bases in $G^n$, exploring asymptotic bounds for $g$, a fixed integer greater than one.
  • It introduces upper bounds for $\nu_g(G^n)$ and $\eta_g(G^n)$, denoting the minimal sizes of $g$-additive and $g$-difference bases respectively.

Mathematical Results

The paper presents several key results:

• Lower Bounds: Establishes trivial lower bounds on $\nu_g(G)$ and $\eta_g(G)$ for abelian groups, utilizing structural properties of these groups.
• Asymptotic Behavior: Investigates $\nu_g(G^n)$ and $\eta_g(G^n)$ as $n \to \infty$. The paper focuses on asymptotically tight lower bounds derived from products of finite abelian groups.
• Existence of Limits: Demonstrates the existence of limits for sequences of normalized minimal basis sizes across families of finite abelian groups.

Implementation Considerations

The mathematical proofs rely heavily on combinatorial constructions and theorems, using intricate counting arguments and integer partitions. For practical implementation in computing or statistical applications:

• Counting Arguments: Techniques for constructing basis sets include careful selections of elements to satisfy conditions $\{ a_i - a_j \mid a_i, a_j \in A \}$ covering target sets densely.
• Trade-off Analysis: As the bases optimize representation redundancy (the $g$ factor), considerations are given for computational overhead in constructing and evaluating potential basis element pairs.

Future Analysis

Potential developments from this research:

• Extended Group Structures: Analyzing similar properties in non-abelian groups or additional nontrivial group extensions.
• Computational Models: Implementing algorithms to visualize or verify these combinatorial bases within larger lattice-arithmetic frameworks.

Conclusion

The paper delivers a powerful combinatorial framework for understanding $g$-additive and $g$-difference bases within Cartesian products of finite abelian groups. The presented asymptotic bounds enrich the field of additive combinatorics by extending classical concepts of Sidon sets and difference bases to higher-order group structures. Future research outside the scope of finite abelian groups -- leveraging these results to tackle more complex combinatorial problems -- could yield significant computational insights.