Periodicity of joint-cotiles

Shrey Sanadhya, PhD
Ben-Gurion University of the Negev

Given a finite set (a tile) F in Z^d, we say that a set A in Z^d is a co-tile of F if the collection of sets F+a, for a in A, form a tiling of Z^d. Namely, these sets are disjoint and their union in Z^d. Given a k-tuple of tiles F_1,...,F_k, we say that A is a joint co-tile of them if A is a co-tile of each F_i. In this talk I will discuss the structure of joint co-tiles in Z^d, particularly, their periodicity. As a result, we extend an old theorem of Newman, stating that every tiling of Z by a single tile is period. If time allows, we will discuss the connections of our new notions to the periodic tiling conjecture, whose Z^2 case was resolved by Bhattacharya, and a counterexample in high dimension was recently given by Greenfeld and Tao. The talk is based on a joint work with Tom Meyerovitch and Yaar Solomon (arXiv:2301.11255).