Dirichlet's Theorem from Diophantine approximation says that for any real number t, there is some v in {1,2,...,n} such that tv lies within 1/(n+1) of an integer. The Lonely Runner Conjecture of Wills and Cusick asserts that the constant 1/(n+1) in this theorem cannot be improved by replacing {1,2,...,n} with a different set of n nonzero real numbers. The conjecture, although now more than 50 years old, remains wide open for n larger than 7. In this talk I will describe a new approach based on the "Lonely Runner spectra" that arise when one considers the "inverse problem" for the Lonely Runner Conjecture. Based on joint work with Vikram Giri and with Vanshika Jain.
