FWF-Kombinatorik u Dynamik v Zif - Combinatorics and dynamics of numeration systems - FWF - Komb. und Dyn. von Ziff.

Systems of numeration give different ways of representing the positive integers as weighted sums; special and classical examples are the decimal expansion (203=2x100+0x10+3x1) or the binary expansion used by arithmetical processors. Alternative expansions allowing either signed digits or more complicated bases than 1,10,100,... have also been studied and used in cryptography, analysis of algorithms, and game theory. The general theory of such systems has its origin in theoretical computer science, where they were studied extensively mostly for speeding up computations. We refer to Knuth's books on the Art of Computer Programming. In the proposed project we take a different point of view, which was initiated by Vershik and Liardet: the representations of the integers are embedded in a topological space. The addition of 1 is used to define a dynamical system called the odometer. Through the somehow intricate structure of the carry propagation this system reflects the combinatorial properties of the underlying number representation. We intend to develop this theory in a general setting and to extend it to other sets such as number fields or polynomial rings. The probabilistic study of classical arithmetical functions has been developed in order to achieve a better understanding of the statistical structure of the prime decomposition of integers. Probabilistic number theory for arithmetical functions related to systems of numeration shall be developed during the course of this project. The space constructed above will play the rle of several compactifications of the integers that have been used for classical arithmetical functions.