On the structure of p-zero-sum free sequences and its application to a variant of Erdos--Ginzburg--Ziv theorem

2005/03/05 by W D Gao, A Panigrahi, R Thangadurai
Mathematics · #math.CO #math.NT #msc:20D60 #msc:11B75

paper · pdf

published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 115, No. 1, February 2005, pp. 67-77 · 11 pages

arxiv created 2005/03/05 · arxiv updated 2009/12/01

Abstract

Let p be any odd prime number. Let k be any positive integer such that 2≤ k≤ [\fracp+13]+1. Let S = (a₁,a₂,...,a_2p-k) be any sequence in \Bbb Z_p such that there is no subsequence of length p of S whose sum is zero in \zp. Then we prove that we can arrange the sequence S as follows: S = (\underbracea, a, ..., a_u \rm times, \underbraceb, b, >..., b_v \rm times, a₁', a₂', >..., a_2p-k-u-v') where u≥ v, u+v≥ 2p-2k+2 and a-b generates \zp. This extends a result in \citegao10 to all primes p and k satisfying (p+1)/4+3≤ k≤ (p+1)/3+1. Also, we prove that if g denotes the number of distinct residue classes modulo p appearing in the sequence S in \zp of length 2p-k (2≤ k≤ [(p+1)/4]+1), and g≥ 2√(2)√(k-2), then there exists a subsequence of S of length p whose sum is zero in \zp.

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