Theoretical Statistics and Mathematics Unit, ISI Delhi

September 3, 2014 (Wednesday) ,
3:30 PM at Webinar

Speaker:
B. K. Moriya,
HRI, Allahabad

Title:
Some weighted zero sum problems

Abstract of Talk

Let $A$ be a non-empty finite subset of integers and $G$ be a finite abelian group. The \textbf{Davenport constant} of $G$ \textbf{with weight} $A$ is defined to be the least $t\in\mathbb{N}$ such that for every sequence $x_1,x_2,\ldots,x_t$ with $x_i\in G$, there exists a non-empty subsequence of length $\ell$, viz. $x_{j_1},\ldots,x_{j_\ell}$ and $a_1,a_2,\ldots,a_\ell\in A$ such that $\sum_{i=1}^\ell a_ix_{j_i}=0,$ and the sequence $x_{j_1},\ldots,x_{j_\ell}$ is called an $A$-weighted zero sum subsequence. Depending on the prescription of the length of an $A$-weighted zero sum subsequence various constants are defined as follows,
\begin{itemize}
\item If $\ell =n=|G|$, we get the \textbf{$A$-weighted EGZ constant}, denoted by $E_A(G)$.
\item If $1\leq \ell \leq \exp(G)$, we get the constant $\eta_A(G).$
\item If $\ell =\exp(G)$, we get the constant $s_A(G)$.
\end{itemize}
These constants are known for very small class of groups. These problems are tackled in two ways. One way is directly try to find out the constant for a given group $G$. Another way is to count the number of sums of the prescribed length of the zero sum free sequences. We will talk about the results related to exact values of these constants and bounds on them. We will also talk about lower bound of the cardinality of $\sum (A)$, set of subset sums of $A$, where $A\subset\Z/p\Z\mbox{ with some constraint and }p$ is an odd prime number.