Disk-like tiles derived from complex bases
For each positive integer \(k\), let \(T_{k}\) be the set of complex numbers representable in the form \(\sum_{j\geq 1}d_{j}(-k+{\mathbf i})^{-j}\) with \( d_{j}\in \{0,1,2,\ldots ,k^{2}\}\). The translates of \(T_{k}\) by Gaussian integers tile the plane, \(T_{k}\) is affinely equivalent to \(\bigcup_{j=1}^{k^{2}}(T_{k}+j)\), and the Hausdorff dimension of the boundary of \(T_{k}\) is a number strictly between \(1\) and \(2\), which has been determined by \textit{W. J. Gilbert} [Can. Math. Bull. 29, 495--500 (1986; Zbl 0564.10007)]. Here the authors show that \(T_{k}\) is homeomorphic to the closed unit disk if and only if \(k\neq 2\). It is noted that this may also be inferred from a result of \textit{C. Bandt} and \textit{Y. Wang} [Discrete Comput. Geom. 26, No. 4, 591--601 (2001; Zbl 1020.52018)], but in a quite different way.
scientific article; zbMATH DE number 2130978