On zero sum subsequences of restricted size
type: Item
instance of: scholarly article
title: On zero sum subsequences of restricted size
zbMATH Open document ID: 1257.11025
author: B. K. Moriya
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2010-12-14T00:00:00Z
review text: Let \(G\) be a finite abelian group. Let \(s(G)\) be the minimal integer \(t\) with the property that any sequence of \(t\) elements in \(G\) contains a subsequence of length \(l=\exp(G)\) with sum zero. Similarly, let \(\eta(G)\) denote the corresponding minimal integer \(t\) where the subsequences are of size \(l \leq \exp(G)\). \textit{N. Alon} and \textit{M. Dubiner} [``A lattice point problem and additive number theory'', Combinatorica 15, No. 3, 301--309 (1995; Zbl 0838.11020)] proved that \(s(C_n^r) \leq c(r) n\) but the determination of \(s(C_n^r)\) (or \(\eta(C_n^r)\)) is a major open problem in this subject. For fixed \(n\) it is expected that \(\eta(C_n^r)=a_{r}(n-1)+1\) and \(s(C_n^r)=(a_{r}+1)(n-1)+1\) holds for some integer \(a_r\). The author proves: If \(\eta(C_m^r)=a_{r}(m-1)+1\) holds, and if \[ n \geq \frac{m^r( c(r)m -a_r(m-1)+m-3 )(m-1)-(m+1)+(m+1)(a_r+1)}{m (m+1)(a_r+1)} \] is a positive integer with \(s(G)=(a_r+1)(n-1)+1\), then \(s(C_{nm}^r)=(a_r+1)(nm-1)+1\).
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11P70
Mathematics Subject Classification ID: 11B50
zbMATH DE Number: 5826102
zbMATH Keywords: zero-sum problems
zbMATH Keywords: Alon-Dubiner constant
reviewed by: Christian Elsholtz
MaRDI profile type: Publication
full work available at URL: s12044-010-0040-1
OpenAlex ID: W2027443207
cites work: A lattice point problem and additive number theory
cites work: Q4342479
cites work: On zero-sum subsequences of restricted size. IV
cites work: Sequences in abelian groups \(G\) of odd order without zero-sum subsequences of length \(\exp(G)\)
cites work: ZERO-SUM PROBLEMS IN FINITE ABELIAN GROUPS AND AFFINE CAPS
cites work: Lower bounds for multidimensional zero sums
cites work: Q5841589
cites work: Q3433845
cites work: Zero-sum problems in finite Abelian groups: a survey
cites work: On long minimal zero sequences in finite abelian groups
cites work: Q4397553
cites work: Q5200682
cites work: Ein Extremalproblem für Gitterpunkte.
cites work: Q3325732
cites work: Q5334400
cites work: Additive group theory—A progress report
cites work: Q4889853
cites work: On Kemnitz' conjecture concerning lattice-points in the plane
DOI: 10.1007/S12044-010-0040-1
Recommended article: On zero-sum subsequences of restricted size
Recommended article: On the Number of Zero-Sum Subsequences of Restricted Size
Recommended article: On zero-sum subsequences of restricted size. II.
Recommended article: On zero-sum subsequences of restricted size. IV
Recommended article: Q5461453
Recommended article: On zero-sum subsequences of prescribed length
Recommended article: On the number of zero-sum subsequences
Recommended article: On the number of zero sum subsequences
Recommended article: On zero-sum subsequences of length not exceeding a given number
Recommended article: On short zero-sum subsequences of zero-sum sequences
description: scientific article; zbMATH DE number 5826102