On Erdős–Ginzburg–Ziv inverse theorems

Abstract

1. Introduction. Let F (G) denote the free abelian monoid over the set G with monoid operation written multiplicatively and given by concatenation, i.e., F (G) consists of all finite sequences over G modulo the equivalence relation allowing terms to be permuted. Despite possible confusion, the elements of F (G) will be referred to simply as sequences, and if indeed order or being infinite are needed in a sequence, it will be explicitly stated when the sequence is first introduced. Now let G be an abelian group of order m ≥ 2. The Erdős-GinzburgZiv theorem states that every sequence in G of length 2m − 1 contains an m-term subsequence with zero sum [5]. There have been many related inverse theorems describing the structure of the sequences S in G with length |S| = m + k, 1 ≤ k ≤ m − 2, not having any m-term subsequence with zero sum. For cyclic groups of order m, the structure of S has been described by several authors: when k = m − 2, by Yuster and Peterson in [15], and by Bialostocki and Dierker in [1]; when k = m − 3, by Flores and Ordaz in [7]; when m − ⌊m/4⌋ − 2 ≤ k ≤ m − 2, by Bialostocki, Dierker, Grynkiewicz, and Lotspeich in [2] (using a related result of Gao from [8]); and when k ≥ ⌈(m − 1)/2⌉, by Chen and Savchev in [3].

On the structure of n-zero-sum free sequences over cyclic groups of order n

“…The Erdős-Ginzburg-Ziv constant s(G) is the smallest integer ∈ N such that every sequence S over G of length |S| ≥ has a zero-sum subsequence of length n. The famous Erdős-Ginzburg-Ziv theorem states that s(G) = 2|G| − 1 if G is cyclic [8]. The Erdős-Ginzburg-Ziv constant has found far reaching generalizations (more information can be found in the surveys [14,21]; for recent progress see [1,2,4,7,9,16,22,31]). …”

Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

“…The Subsum Kneser's Theorem can alternatively be derived as a special case of the DeVos-Goddyn-Mohar Theorem [9] [37]. Theorem C, and the more general Theorem A, have found numerous use in problems regarding sequence subsums [5] [15] [18] [19] [23] [25] [27] [28] [29] [30] [31] [32] [33] [34] [36], extending, complementing or resolving questions of established interest [2] [3] [4] [6] [7] [8] [10] [11] [12] [13] [14] [17] [39] [40] [41] [44] [46] [50].…”