Title: A lonely weak tile
Abstract: Let A be a bounded measurable subset of R^d. Fuglede conjectured that A tiles R^d by translations if and only if it is a spectral set. Though the conjecture is false in full generality, it has been confirmed for convex sets. A key ingredient of the proof of the latter result is the notion of weak tiling. It has been proved by Lev and Matolcsi that every spectral set tiles its complement weakly with a suitable Borel measure. In my talk I will answer positively a question raised by Kolountzakis, Lev and Matolcsi whether there exists a set in R^d tiling its complement weakly but is neither spectral nor a tile. Based on joint work with Gergo Kiss, Mate Matolsci and Gabor Somlai