Erdos' minimum overlap problem

Abstract: In 1955 Erd\H{o}s posed the following problem. Let $n$ be a positive integer and $A,B \subset [2n A weighted generalization of classical zero-sum constants was introduced by Adhikari {\it et al.} in 2006 and has been an active area of research since then. In the last fifteen years, weighted zero-sum constants for $\mathbb {Z}_n$ with several interesting weight sets have been found. In this talk, we take up the problem of determining the exact values and providing bounds of the weighted Davenport constant of $\mathbb {Z}_n$ with some new weight sets.

Next, we consider a weighted generalization of the {\it the Erd\H{o}s-Ginzburg-Ziv constant}. Let $G$ be a finite abelian group with $\exp(G)=n$. For a positive integer $k$ and a non-empty subset $A$ of $[1, n-1]$, the arithmetical invariant $\mathsf s_{kn,A}(G)$ is defined to be the least positive integer $t$ such that any sequence $S$ of $t$ elements in $G$ has an $A$-{\it weighted zero-sum subsequence} of length $kn$. We give the exact value of $\mathsf s_{kq,A}(G)$, for integers $k\geq 2$ and $A=\{1,2\}$, where $G$ is an abelian $p$-group with $rank(G)\leq 4$, $p$ is an odd prime and $exp(G)=q$. Our method consists of a modification of a polynomial method of R\'onyai.

Lastly, we consider the questions regarding inverse problems for the weighted zero-sum constants of $\mathbb {Z}_n$. An inverse problem is the problem of characterizing all the weighted {\it zero-sum free sequences} over $\mathbb {Z}_n$ of specific lengths for the particular weight sets under consideration.

This work was joint with Sukumar Das Adhikari and partly with Md Ibrahim Molla and Subha Sarkar. ]$ be a partition of $[2n]$ such that $|A|=|B| = n$. For any such partition and integer $-2n0.379$.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)