Probability and Mathematical Physics Seminar

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Covering systems of congruences

Speaker: Bob Hough, Stony Brook

Location: Warren Weaver Hall 1302

Date: Friday, October 14, 2022, 3 p.m.

Synopsis:

A distinct covering system of congruences is a list of congruences \[ a_i \bmod m_i, \qquad i = 1, 2, ..., k \] whose union is the integers. Erd\H{o}s asked if the least modulus $m_1$ of a distinct covering system of congruences can be arbitrarily large (the minimum modulus problem for covering systems, $1000) and if there exist distinct covering systems of congruences all of whose moduli are odd (the odd problem for covering systems, $25). I'll discuss my proof of a negative answer to the minimum modulus problem, and a quantitative refinement with Pace Nielsen that proves that any distinct covering system of congruences has a modulus divisible by either 2 or 3. The proofs use the probabilistic method and in particular use a sequence of pseudorandom probability measures adapted to the covering process. Time permitting, I may briefly discuss a reformulation of our method due to Balister, Bollob\'{a}s, Morris, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering system of congruences has one modulus that divides another) and gives a negative answer to the square-free version of the odd problem.