The present project aims at gaining substantial results on the interaction of different numeration systems. In
particular, the proposer wants to analyze positional number systems, like radix systems, U-scales and beta
expansions from different points of view. These may be dynamical systems, statistics and distribution and ergodic
theory. Some of the mentioned systems have already been the object of intensive studies, like the dynamical system
of beta expansions or the asymptotic distribution of additive functions in radix expansions. Nevertheless one does
not know much about their interaction and interchangeability. Therefore the main goal of this project is to
investigate their connections and similarities as well as their disconnections and differences between these different
notions of numeration systems. The methods, which are used for the investigation, originate from different areas of
analysis and probability theory such as Fourier analysis, analysis of algorithms, analytic combinatorics, automata
theory, Markov models, martingale theory, the analysis of weakly dependent random variables and ergodic theory.
The study of a single point of view is quite often based on the usage of methods, which were especially developed
and designed to fit the requirements of a precise numeration system. Therefore the central idea of the project is to
find new points of interactions of these methods and how they may be applied in more generality to obtain new
perspectives in different numeration systems. This should lead to a more complete view on the available methods,
their applications and their interactions. Since numeration systems have connections and relations to different areas
such as combinatorics, Diophantine approximation, ergodic theory and computer science, the applicant looks
forward to obtain considerable results for these objectives too.