A classical result by Leptin-Müller and Lagarias-Kolountzakis shows that a positive, compactly supported function on the real line can tile the real line by translations only if the translation set is a finite union of periodic sets. The aim of the presentation is to provide an example showing that this is not the case if the function is allowed to have unbounded support.
Reference:
Kolountzakis, M., Lev, N. On non-periodic tilings of the real line by a function. Int. Math. Res. Not. IMRN, No. 15, 4588-4601 (2016).