Improved Bounds Towards the Lonely Runner Conjecture

The Lonely Runner Conjecture, due to Wills and Cusick, asserts that if n runners with distinct constant speeds run around a unit length track, all starting at a common point, then each runner is at some moment separated by a distance of at least 1/n from every other runner. 

A weaker lower bound of 1/2n follows from the so-called trivial union bound. Subsequent work improved this to bounds of the form 1/(2n) + c/n^2, for various constants c > 0. Tao later strengthened this by a logarithmic factor, obtaining the previous record bound 1/(2n) + (log n)/n^2. In this talk, we discuss a recent polynomial improvement.