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

Abstract

Let p be any odd prime number. Let k be any positive integer such that 2 ≤ k ≤ p+1 3 + 1. Let S = (a 1 , a 2 , . . . , a 2p−k ) be any sequence in Z p such that there is no subsequence of length p of S whose sum is zero in Z p . Then we prove that we can arrange the sequence S as follows:This extends a result in [13] 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 Z p 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 Z p .

A Structural Approach to Subset-Sum Problems

“…Few years ago, Gao, Panigrahi, and Thangdurai [43] proved a similar statement under the stronger assumption that |A| ≥ 3p/2.…”

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

“…Gao [12] characterized the n-zero-sum free sequences of length roughly greater than 7n 4 . Gao, Panigrahi and Thangadurai [18] considered the same question for sequences of length roughly greater than 5n 3 providing that n is prime. The best result so far is achieved by Savchev and Chen [30] in 2008.…”

On Erdős–Ginzburg–Ziv inverse theorems

“…Problem 1 ( [10,12]). For an abelian group G of order m ≥ 2 and a positive integer k, determine the exact value or a bound of h(G, k) = min{h(S) : S ∈ F (G) with |S| = |G| + k and 0 ∈ Σ |G| (S)}.…”