Generalizations of some Zero Sum Theorems
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Given an abelian group 𝐺 of order 𝑛, and a finite non-empty subset 𝐴 of integers, the Davenport constant of 𝐺 with weight 𝐴, denoted by $D_A(G)$, is defined to be the least positive integer 𝑡 such that, for every sequence $(x_1,\ldots,x_t)$ with $x_i\in G$, there exists a non-empty subsequence $(x_{j_1},\ldots,x_{j_l})$ and $a_i\in A$ such that $\sum^l_{i=1}a_ix_{j_i}=0$. Similarly, for an abelian group 𝐺 of order $n,E_A(G)$ is defined to be the least positive integer 𝑡 such that every sequence over 𝐺 of length 𝑡 contains a subsequence $(x_{j_1},\ldots,x_{j_n})$ such that $\sum^n_{i=1}a_ix_{j_i}=0$, for some $a_i\in A$. When 𝐺 is of order 𝑛, one considers 𝐴 to be a non-empty subset of $\{1,\ldots,n-1\}$. If 𝐺 is the cyclic group $\mathbb{Z}/n\mathbb{Z}$, we denote $E_A(G)$ and $D_A(G)$ by $E_A(n)$ and $D_A(n)$ respectively.
In this note, we extend some results of Adhikari et al(Integers 8(2008) Article A52) and determine bounds for $D_{R_n}(n)$ and $E_{R_n}(n)$, where $R_n=\{x^2:x\in(\mathbb{Z}/n\mathbb{Z})^∗\}$. We follow some lines of argument from Adhikari et al(Integers 8 (2008) Article A52) and use a recent result of Yuan and Zeng (European J. Combinatorics 31 (2010) 677–680), a theorem due to Chowla (Proc. Indian Acad. Sci. (Math. Sci.) 2 (1935) 242–243) and Kneser’s theorem (Math. Z.58(1953) 459–484;66(1956) 88–110;61(1955) 429–434).
M N Chintamani^{1} ^{} B K Moriya^{1} ^{}
Volume 133, 2023
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