Characterization of a class of planar self-affine tile digit sets

Abstract:

We call a finite set ${\mathcal {D}}\subset {\Bbb Z}^s$ a (self-affine) tile digit set with respect to an expanding integral matrix $\textbf {A}$ if the self-affine set $T(\textbf {A}, \mathcal {D})$ is a tile in ${\Bbb R}^s$. It has been a widely open problem to characterize the tile digit sets for a given $\textbf {A}$. While there are substantial investigations on ${\Bbb R}$, there is no result on ${\Bbb R}^s$ other than the case where $|\det \textbf {A}| =p$ with $p$ a prime. In this paper, we make an initiation to study a basic case $\textbf {A} = p\textbf {I}_2$ in ${\Bbb R}^2$. We characterize the tile digit sets by making use of the zeros of the mask polynomial of ${\mathcal {D}}$ associated with a tile criterion of Kenyon [Self-replicating tilings, Contemp. Math., vol. 135, Amer. Math. Soc., Providence, RI, 1992, pp. 239–263], together with a recent result of Iosevich et al. on factorization of sets in ${\Bbb Z}_p \times {\Bbb Z}_p$ [Anal. PDE 10 (2017), no. 4, 757–764].