Dynamics, geometry, and arithmetic of Numeration

The representation of numbers and counting is as old as human civilization and has fascinated mankind for thousands of years. The progress of mathematics and, later on, of computer science made it necessary to come up with generalized and more sophisticated ways to represent numbers and entailed the need of a deeper understanding of such representations. The present project uses methods from different fields of mathematics, like dynamical systems theory, fractal geometry, and number theory to enhance and deepen the understanding of number representations. In the present project so-called continued fraction expansions are studied. Such representations are necessary when one wishes to approximate an irrational number by a rational number in an optimal way. As already Emil Artin observed some eighty years ago, these representations have striking connections to the hyperbolic geometry of the upper half plane and can be interpreted as certain sections of a geodesic flow on this half plane. This connection helped to gain new understanding of continued fraction expansions. In the present project, we consider generalizations of continued fraction expansions and relate them to geodesic flows in higher dimensional spaces. In this context many new phenomena occur and fractal sets that contain information on these continued fraction expansions come up in a natural way. Another task we are concerned with is related to an almost 50 years old conjecture in number theory that is easy to state but extremely hard to prove: in 1968 Gelfond asked if there are infinitely many prime numbers whose sum of digits is even (e.g. 19 is a prime with even sum of digits 1+9 = 10). Using sophisticated methods from the theory of exponential sums, Mauduit and Rivat were able to prove this conjecture five years ago. However, there are natural and important ways to represent numbers and in this context the conjecture is still open. Recently, together with M. Madritsch, the proposer found a new result on exponential sums that could shed some light on the representation of numbers as sums of Fibonacci numbers (so-called Zeckendorf expansions): within the present project we would like to use this result in order to study the representation of primes in Zeckendorf expansions. In recent decades a big variety of number systems have been invented. Although they are defined in rather different ways, the proposer and his co-authors invented shift radix systems. These simple dynamical systems contain many seemingly different kinds of number systems as special cases. It is our aim to study these expansions as well as the fractal sets that are related to them in a natural way.

In the framework of the present project new results on number systems could be achieved. These objects play an important role in Mathematics as well as in Theoretical Computer Science. A key feature of the present project was the investigation of number systems in the field of Dynamical Systems and Fractal Geometry.