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- W3207680071 abstract "We consider digit systems $(A,mathcal{D})$, where $ A in mathbb{Q}^{ntimes n}$ is an expanding matrix and the digit set $mathcal{D}$ is a suitable subset of $mathbb{Q}^n$. To such a system, we associate a self-affine set $mathcal{F} = mathcal{F}(A,mathcal{D})$ that lives in a certain representation space $mathbb{K}$. If $A$ is an integer matrix, then $mathbb{K} = mathbb{R}^n$, while in the general rational case $mathbb{K}$ contains an additional solenoidal factor. We give a criterion for $mathcal{F}$ to have positive Haar measure, i.e., for being a rational self-affine tile. We study topological properties of $mathcal{F}$ and prove some tiling theorems. Our setting is very general in the sense that we allow $(A,mathcal{D})$ to be a nonstandard digit system. A standard digit system $(A,mathcal{D})$ is one in which we require $mathcal{D}$ to be a complete system of residue class representatives w.r.t. a certain naturally chosen residue class ring. Our tools comprise the Frobenius normal form and character theory of locally compact abelian groups." @default.
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- W3207680071 date "2021-10-18" @default.
- W3207680071 modified "2023-10-06" @default.
- W3207680071 title "Rational self-affine tiles associated to standard and nonstandard digit systems" @default.
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