On the number of m-term zero-sum subsequences

Abstract

A sequence S of terms from an abelian group is zero-sum if the sum of the terms of S is zero. In 1961 Erdős, Ginzburg and Ziv proved that any sequence of 2m− 1 terms from an abelian group of order m contains an m-term zero-sum subsequence [10]. This sparked a flurry of generalizations, variations and extensions [1] [3] [7] [8] [11] [13] [14] [15] [16] [17] [18] [22] [26] [27] [28] [37]. Since a sequence from the cyclic group Z/mZ consisting of only 0’s and 1’s has its m-term zerosum subsequences in exact correspondence with its m-term monochromatic subsequences, then the Erdős-Ginzburg-Ziv Theorem can be viewed as a generalization of the pigeonhole principle for m pigeons and 2 boxes. In essence, the Erdős-Ginzburg-Ziv Theorem expresses the idea that often the best way to avoid zero-sums is to consider sequences with very few distinct terms. For sequences whose length is greater than 2m−1, a natural question to ask is how many m-term zero-sum subsequences can one expect. If the sequence S has length n and consists of at most two distinct terms, then there will be at least (dn 2 e m ) + (bn 2 c m ) m-term monochromatic subsequences. Thus ∗AMS Subject Classification: 11B75 (11B50)