On the Structure of 𝑝-Zero-Sum Free Sequences and its Application to a Variant of Erdös–Ginzburg–Ziv Theorem

[摘要] Let 𝑝 be any odd prime number. Let 𝑘 be any positive integer such that $2 ≤ k ≤ left[frac{p+1}{3}ight]+1$. Let $S =(a_1,a_2,ldots,a_{2p−k})$ be any sequence in $mathbb{Z}_p$ such that there is no subsequence of length 𝑝 of 𝑆 whose sum is zero in $mathbb{Z}_p$. Then we prove that we can arrange the sequence 𝑆 as follows:$$S=(underset{uext{times}}{underbrace{a,a,ldots,a}},underset{vext{times}}{underbrace{b,b,ldots,b}},{a'}_1,{a'}_2,ldots,{a'}_{2p-k-u-v})$$where $u≥ v, u + v ≥ 2p − 2k + 2$ and 𝑎 − 𝑏 generates $mathbb{Z}_p$. 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 $mathbb{Z}_p$ of length $2p − k(2≤ k≤ [(p + 1)/4]+1)$, and $g≥ 2sqrt{2}sqrt{k - 2}$, then there exists a subsequence of 𝑆 of length 𝑝 whose sum is zero in $mathbb{Z}_p$.

[学科分类] 数学（综合）

[关键词] Sequences;zero-sum problems;zero-free;Erdös–Ginzburg–Ziv theorem.