The topological structure of fractal tilings generated by quadratic number systems

Let @a be a root of an irreducible quadratic polynomial x^2 + Ax + B with integer coefficients A, B and assume that @a forms a canonical number system, i.e., each x @? @?[@a] admits a representation of the shape x=a"0+a"1@a+...+a"h@a^h,with a"i @? {0, 1,...,|B| - 1}. It is possible to associate a tiling to such a number system in a natural way. If 2A = B + 3, then the topological structure of the tiles is quite involved. In this case, we prove that the interior of a tile is disconnected. Furthermore, we are able to construct finite labelled directed graphs which allow to determine the set of ''neighbours'' of a given tile T, i.e., the set of all tiles which have nonempty intersection with T. In a next step, we give the structure of the set of points, in which T coincides with L other tiles. In this paper, we use two different approaches: geometry of numbers and finite automata theory. Each of these approaches has its advantages and emphasizes different properities of the tiling. In particular, the conjecture in [1], that for A 0 and 2A < B + 3 there exist exactly six points where T coincides with two other tiles, is solved in these two ways in Theorems 6.6 and 10.1.