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
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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.
Watch the video:
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.