Modified Erd\H{o}s-Ginzburg-Ziv constants for $\mathbb{Z}_2^d$

TL;DR

This paper determines exact values of the modified Erdős-Ginzburg-Ziv constants for certain finite abelian groups, specifically for groups of the form 2^d, advancing understanding of zero-sum sequences in additive combinatorics.

Contribution

It provides explicit calculations of 2^d for specific parameters, filling gaps in the knowledge of zero-sum problems in finite abelian groups.

Findings

Exact values of 2^d for d  2k+1

Advances in zero-sum sequence theory for 2^d

Improved bounds for Erds-Ginzburg-Ziv constants

Abstract

Let $G$ be a finite abelian group written additively, and let $r$ be a multiple of its exponent. The modified Erd\H{o}s-Ginzburg-Ziv constant $\mathsf{s}_r'(G)$ is the smallest integer $s$ such that every zero-sum sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We find exact values of $\mathsf{s}_{2k}'(\mathbb{Z}_2^d)$ for $d \leq 2k+1$.