Rob Morris "Erdős covering systems"
Rob Morris from IMPA will give the talk "Erdős covering systems" on the labs' Big Seminar.
Password: first 6 decimal places of $\pi$ after the decimal point
You can also write to Alexander Polyanskii (alexander.polyanskii@yandex.ru) or to Maksim Zhukovskii (zhukmax@gmail.com) if you want to be added to mailing list.
Abstract:
A covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called "minimum modulus problem" was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$.
In this talk I will present a variant of Hough's method, which turns out to be both simpler and more powerful. In particular, I will sketch a short proof of Hough's theorem, and discuss several further applications. I will also discuss a related result, proved using a different method, about the number of minimal covering systems.
Joint work with Paul Balister, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.
Everyone is invited to attend. The language of the lecture is English. The event is aimed at master and graduate students, as well as researchers in the field of combinatorics.