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

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/115/01/0067-0077

    • Keywords

       

      Sequences; zero-sum problems; zero-free; Erdös-Ginzburg-Ziv theorem

    • Abstract

       

      Letp be any odd prime number. Letk be any positive integer such that $$2 \leqslant k \leqslant \left[ {\frac{{p + 1}}{3}} \right] + 1$$. LetS = (a1,a2,...,a2p−k) be any sequence in ℤp such that there is no subsequence of lengthp of S whose sum is zero in ℤp. Then we prove that we can arrange the sequence S as follows: $$S = (\underbrace {a,a,...,a,}_{u times}\underbrace {b,b,...,b,}_{v times}a'_1 ,a'_2 ,...,a'_{2p - k - u - v} )$$whereuv,u +v ≥ 2p - 2k + 2 anda -b generates ℤp. This extends a result in [13] to all primesp andk satisfying (p + 1)/4 + 3 ≤k ≤ (p + 1)/3 + 1. Also, we prove that ifg denotes the number of distinct residue classes modulop appearing in the sequenceS in ℤp of length 2p -k (2≤k ≤ [(p + 1)/4]+1), and $$g \geqslant 2\sqrt 2 \sqrt {k - 2} $$, then there exists a subsequence of S of lengthp whose sum is zero in ℤp.

    • Author Affiliations

       

      W D Gao1 A Panigrahi1 2 R Thangadurai1 2

      1. Department of Computer Science and Technology, University of Petroleum, Changping Shuiku Road, Beijing - 102200, China
      2. School of Mathematics, Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad - 211 019, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2017-2019 Indian Academy of Sciences, Bengaluru.