Zero-sum problems with congruence conditions

2010/07/01 by Alfred Geroldinger, Geroldinger, Alfred, David J. Grynkiewicz +3 · 1 citation
Mathematics · #11B30 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:11B30

paper · pdf · doi:10.48550/arxiv.1007.0251

arxiv created 2010/07/01 · arxiv updated 2010/07/05

Abstract

For a finite abelian group G and a positive integer d, let \mathsf s_{d \mathbb N} (G) denote the smallest integer ℓ ∈ \mathbb N₀ such that every sequence S over G of length |S| ≥ ℓ has a nonempty zero-sum subsequence T of length |T| ≡ 0 \mod d. We determine \mathsf s_{d \mathbb N} (G) for all d≥ 1 when G has rank at most two and, under mild conditions on d, also obtain precise values in the case of p-groups. In the same spirit, we obtain new upper bounds for the Erd\H os--Ginzburg--Ziv constant provided that, for the p-subgroups G_p of G, the Davenport constant \mathsf D (G_p) is bounded above by 2 exp (G_p)-1. This generalizes former results for groups of rank two.

Cited by

• On zero-sum subsequences in a finite abelian group of length not exceeding a given number

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