• A Variant of Davenport's Constant

    • Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/117/02/0147-0158

    • Keywords

       

      Davenport’s constant; zero-sum problems; abelian groups.

    • Abstract

       

      Let 𝑝 be a prime number. Let 𝐺 be a finite abelian 𝑝-group of exponent 𝑛 (written additively) and 𝐴 be a non-empty subset of $]n[:=\{1,2,\ldots,n\}$ such that elements of 𝐴 are incongruent modulo 𝑝 and non-zero modulo 𝑝. Let $k \geq D(G)/|A|$ be any integer where $D(G)$ denotes the well-known Davenport’s constant. In this article, we prove that for any sequence $g_1,g_2,\ldots,g_k$ (not necessarily distinct) in 𝐺, one can always extract a subsequence $g_{i_1},g_{i_2},\ldots,g_{i_l}$ with $1 \leq l \leq k$ such that

      $$\sum\limits_{j=1}^l a_j g_{i_j}=0 \text{in} G,$$

      where $a_j\in A$ for all 𝑗. We provide examples where this bound cannot be improved. Furthermore, for the cyclic groups, we prove some sharp results in this direction. In the last section, we explore the relation between this problem and a similar problem with prescribed length. The proof of Theorem 1 uses group-algebra techniques, while for the other theorems, we use elementary number theory techniques.

    • Author Affiliations

       

      R Thangadurai1

      1. 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.