Yaar Solomon

Periodicity of joint co-tiles in Z^d.

The periodic tiling conjecture in Z^d asserts that if a tile (=finite set) tiles Z^d, then it must also tile it periodically. In dimension d=1, an old theorem of Newman shows an even stronger assertion, which is that every tiling of Z is itself periodic. Very recently, Greenfeld and Tao showed that the periodic tiling conjecture is false in large enough dimension d. On the other hand, Bhattacharya recently proved for d=2 that the periodic tiling conjecture is true. In this talk, after giving all the definitions and background, I'll explain how (in spite of Greenfeld-Tao counterexample) both Newman's and Bhattacharya's theorems can be extended to any dimension d, with a slightly different statement and setup.

This talk is based on a joint work with Tom Meyerovitch and Shrey Sanadhya.