Cyclotomic Aperiodic Substitution Tilings with Dense Tile Orientations

The class of Cyclotomic Aperiodic Substitution Tilings (CAST) whose vertices are supported by the $2n$-th cyclotomic field $\mathbb{Q}\left({\zeta }_{2n}\right)$ is extended to cases with Dense Tile Orientations (DTO). It is shown that every CAST with DTO has an inflation multiplier $\eta$ with irrational argument so that $\frac{k\pi }{2n}\mathrm{\ne }\arg \left(\eta \right)\mathrm{\notin }\mathbb{\pi }\mathbb{Q}$. The minimal inflation multiplier ${\eta }_{min\mathrm{.}irr\mathrm{.}}$ is discussed for $n\geqq 2$. Examples of CASTs with DTO, minimal inflation multiplier ${\eta }_{min\mathrm{.}irr\mathrm{.}}$ and individual dihedral symmetry $D_{2n}$ are introduced for $n\in \{2,3,4,5,6,7\}$. The examples for $n\in \{2,3,4,5,6\}$ also yield finite local complexity (FLC).