Number Systems, Tilings, and Seminumerical Algorithms

Number systems are familiar mathematical objects having their initial origin in problems of everyday life. Starting with the well-known q-adic number systems, many generalized notions of number systems have been studied in literature. In the present project we want to continue the study of properties of various generalized number systems like canonical number systems and beta expansions. It turns out, that this requires methods from several branches of mathematics like number theory, topology, complex analysis, symbolic computation, automata theory and fractal geometry. It is a difficult problem to characterize all numbers which can serve as base numbers of canonical number systems and beta expansions. Using a generalization which unifies both of these notions of number systems we want to achieve new results on the characterization of their bases. Recent work on this problem suggests that this requires the study of sets of fractal-like shape. To canonical number systems and to beta-expansions we can associate fractal tiles in a natural way. In general, the tiles associated to canonical number systems are better understood than the tiles associated to beta expansions. We want to study a class of tiles which contains all these tiles as subclasses. From this we hope to be able to transfer results from the canonical number systems setting to tiles associated to beta expansions. The tiles mentioned before have fractal structure in general. We want to study these tiles with respect to their topological properties. In particular, we want to give criteria for the non-triviality of their fundamental group. In connection with the base characterization as well as in the study of the topology of tiles certain seminumerical algrorithms play an important role. These algorithms should be implemented with help of a computer algebra system. Another topic is the sum of digits function which can be associated to generalized number systems. We want to carry over results which are known for the q-adic sum of digits function to this general setting. Some results have already been generalized. In the proofs of these results the properties of the tiles associated to generalized number systems play an important role. Other tools that we are going to use here are exponential sum estimates and the circle method.