Title. Erdos Covering Systems

Abstract. Since their introduction by Erdos in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. In 1950, Erdos asked if there exist covering systems with distinct arbitrary large moduli. In 1965, Erdos and Selfridge asked if there exist covering systems with distinct odd moduli. In 1967, Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. In 2015, Hough resolved Erdos' problem showing that a finite collection of arithmetic progressions with distinct sufficiently large moduli does not cover the integers. We established a quantitative version of Hough's theorem estimating the density of the uncovered set, thus answering a question posed by Filaseta, Ford, Konyagin, Pomerance and Yu from 2007. Additionally, we resolved the Erdos-Selfridge problem in the square free case as well as Schinzel's conjecture in full generality. In this talk, we discuss these results and present a gentle exposition of the methods used. This talk is based on joint work with Paul Balister, Bela Bollobas, Rob Morris and Julian Sahasrabudhe.

The Brunn-Minkowski inequality is a fundamental geometric inequality, closely related to the isoperimetric inequality. It states that for (open) sets $A$ and $B$ in $\mathbb{R}^d$, we have $|A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}$. Here $A+B=\{a+b: a \in A, b \in B\}$. Equality holds if and only if $A$ and $B$ are convex and homothetic sets (one is a dilation of the other) in $\mathbb{R}^d$. The stability of the Brunn-Minkowski inequality is the principle that if we are close to equality, then A and B must be close to being convex and homothetic. We prove a sharp stability result for the Brunn-Minkowski inequality, establishing the exact dependency between the two notions of closeness, thus concluding a long line of research on this problem. This is joint work with Alessio Figalli and Peter van Hintum.