• B K Moriya

Articles written in Proceedings – Mathematical Sciences

• On Zero Sum Subsequences of Restricted Size

Let 𝐺 be a finite abelian group with $\exp(G)=e$. Let $s(G)$ be the minimal integer 𝑡 with the property that any sequence of 𝑡 elements in 𝐺 contains an 𝑒-term subsequence with sum zero. Let $n, m$ and 𝑟 be positive integers and $m\geq 3$. Furthermore, $\eta(C^r_m)=a_r(m-1)+1$, for some constant $a_r$ depending on 𝑟 and 𝑛 is a fixed positive integer such that

$$n\geq\frac{m^r(c(r)m-a_r(m-1)+m-3)(m-1)-(m+1)+(m+1)(a_r+1)}{m(m+1)(a_r+1)}$$

and $s(C^r_n)=(a_r+1)(n-1)+1$. In the above lower bound on $n,c(r)$ is the Alon-Dubiner constant. Then $s(C^r_{nm})=(a_r+1)(nm-1)+1$.

• Generalizations of some Zero Sum Theorems

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).

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019