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

Li-Xiang An Central China Normal University Ka-Sing Lau The Chinese University of HongKong&Central China Normal University

Number Theory mathscidoc:1701.24009

We call a finite set ${\mathcal{D}}\subset {\Bbb Z}^s$ a {\it (self-affine) tile digit set} with respect to an expanding integral matrix ${\bf A}$ if the self-affine set $T({\bf A}, \D)$ is a tile in ${\Bbb R}^s$. It has been a widely open problem to characterize the tile digit sets for a given ${\bf A}$. While there are substantial investigations on ${\Bbb R}$, there is no result on ${\Bbb R}^s$ other than the case where $|\det {\bf A}| =p$ with $p$ a prime. In this paper, we make an initiation to study a basic case ${\bf A} = p{\bf I}_2$ 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 [K], together with a recent result of Iosevich {\it et al} on factorization of sets in ${\Bbb Z}_p \times {\Bbb Z}_p$ [IMP].
Digit sets, direct summands, mask polynomials, modulus, prime number, spectral sets, self-affine tiles, tile digit sets,zeros
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@inproceedings{li-xiangcharacterization,
  title={Characterization of a class of planar self-affine tile digit sets},
  author={Li-Xiang An, and Ka-Sing Lau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170113170237490514101},
}
Li-Xiang An, and Ka-Sing Lau. Characterization of a class of planar self-affine tile digit sets. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170113170237490514101.
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