Self-affine tiles in \(\mathbb{R}^n\) (Q1923995)

A set \(T\) in \(\mathbb{R}^n\) is called a self-affine tile, if it is compact, with non-empty interior, and if there are essentially disjoint translates of \(T\) whose union is an affine image of \(T\); i.e. there exists a finite set \(D=\{{\underset \sim d}_1, \dots, {\underset \sim d}_m\}\) of ``digits'', and a linear transformation \(A\) with all eigenvalues of absolute value greater than 1, such that \(A(T)= \bigcup^m_{i=1} (T+ {\underset \sim d}_i)\). This is a generalization of self-similar tiles, where \(A\) is a similarity. For a self-affine tile, \(D\) and \(A\) are not unique. Conversely, given \(D\) and \(A\) (such that \(|\text{det} A|=m)\) there is a unique compact set \(T=\{\sum^\infty_{j=1} A^{-j} {\underset \sim d}_{i_j}\): each \({\underset \sim d}_{i_j}\in D\}\). A first theorem gives conditions on \(A\) and \(D\) for \(T\) to have non-empty interior; then \(T\) is the self-affine tile associated with \(A\) and \(D\). A second theorem reproves that every self-affine tile gives a tiling of \(\mathbb{R}^n\) by translations, it also shows that every self-affine tile can be used as a prototile for a self-replicating of \(\mathbb{R}^n\) in the sense of Kenyon. The third theorem adds a converse to Kenyon's rigidity theorem concerning quasiperiodic self-replicating tilings. The paper closes with some open problems and conjectures.