Covers of the Integers by Residue Classes and their Extensions to Groups

2:00pm, Tuesday 11th February

Abstract

A system $A=\{a_s+n_s\mathbb Z\}_{s=1}^k$ of $k$ residue classes is called a cover of $\mathbb Z$.

if any integer belongs to one of the $k$ residue classes. This concept was introduced by P. Erdős in the 1950s. Erdős ever conjectured that $A$ is a cover of $\mathbb Z$ whenever it covers $1,\ldots,2^k$.

In this talk we introduce some basic results on covers of $\mathbb Z$ as well as their elegant proofs.

We will also talk about covers of groups by finitely many cosets, give a proof of the Neumann-Tomkinson theorem, and introduce progress on the Herzog-Sch\"onheim conjecture and the speaker's conjecture on disjoint cosets.