• A Variant of Davenport's Constant

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    • 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

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      Posted on July 25, 2019

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