A Variant of Davenport's Constant
Click here to view fulltext PDF
Permanent link:
https://www.ias.ac.in/article/fulltext/pmsc/117/02/0147-0158
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.
R Thangadurai^{1} ^{}
Volume 131, 2021
All articles
Continuous Article Publishing mode
Click here for Editorial Note on CAP Mode
© 2021-2022 Indian Academy of Sciences, Bengaluru.