Date:
Tue, 19/12/202312:00-13:00
Abstract: Given a finite set (a tile) $F \subset \mathbb{Z}^d$, we say that a set $A \subset \mathbb{Z}^d$ is a co-tile of $F$ if the collection of sets $F+a$, for $a \in A$, forms a tiling of $\mathbb{Z}^d$. For finitely many tiles $F_1,...,F_k$, we say that $A$ is a joint co-tile if $A$ is a co-tile of each $F_i$.
In this talk, we will discuss the structure of joint co-tiles in $\mathbb{Z}^d$, particularly their periodicity. We will discuss the connections of this notion to the periodic tiling conjecture (PTC), whose $\mathbb{Z}^2$ case was resolved by Bhattacharya, and a counterexample in high dimension was recently given by Greenfeld-Tao. Our setup extends a theorem of Newman (every co-tile in $\mathbb{Z}$ is periodic) to higher dimensions. It provides characterization for the periodicity of a co-tile for all $d>2$. This is joint work with Tom Meyerovitch and Yaar Solomon [arXiv:2301.11255].
In this talk, we will discuss the structure of joint co-tiles in $\mathbb{Z}^d$, particularly their periodicity. We will discuss the connections of this notion to the periodic tiling conjecture (PTC), whose $\mathbb{Z}^2$ case was resolved by Bhattacharya, and a counterexample in high dimension was recently given by Greenfeld-Tao. Our setup extends a theorem of Newman (every co-tile in $\mathbb{Z}$ is periodic) to higher dimensions. It provides characterization for the periodicity of a co-tile for all $d>2$. This is joint work with Tom Meyerovitch and Yaar Solomon [arXiv:2301.11255].