Expansions, dyanmical Systems and Tilings – EST

In everyday life we need to represent numbers. Besides the omnipresent
decimal expansion, we not only use binary, octal and hexadecimal
representations but also signed binary or Zeckendorf expansion and
continued fractions. An understanding of uniqueness, how to perform
mathematical operations and approximation of values in real life is
necessary in order to properly use these more advanced representations
of numbers. In the present project we consider these expansions from
different points of view. Any positional numeration system describes
an additive structure and therefore multiplicative structures like the
primes should be indistinguishable from random structures. In
particular, the digits of primes or polynomial values should
distribute equally. The Zeckendorf expansion is only an example of a
larger class of linear recurrent number systems. Each of these systems
is connected with some beta-expansion, where beta is the dominant root
of the characteristic polynomial. Many properties of linear recurrent
number systems have their correspondence in beta-expansion and vice
versa. One part of the present project investigates these
correspondences. The first task is the search for multidimensional
low-discrepancy sequences, so called Halton sequences, that are
defined over linear recurrent number systems. The underlying
transformation is ergodic and for the calculation of its invariant
measure we have to check whether there occurs matching. This means
that the orbit of the left- and rightmost endpoint of the fundamental
interval coincide after a certain number of steps, which we perform in
Task 2. A useful tool for the analysis of linear recurrent number
systems of higher degree is the associated Rauzy fractal. In the third
task we want to consider prime elements and their distribution on the
Rauzy fractal. This involves an analysis of the corresponding ring of
integers. The fourth task considers the distribution of the digits of
primes in linear recurrent number systems. For this task we need a
combination of analytic and ergodic methods. The second part considers
a different expansion: the continued fraction expansion. In the
classical case it provides us with best approximations for reals. A
multidimensional variant has been the interest of research for almost
one-hundred years but still no satisfying generalisation has been
found. In Task 5 we investigate the Hurwitz continued fraction
operator and its properties for the approximation of complex
numbers. The last task deals with multidimensional Diophantine
approximation and minimal vectors. Diophantine approximation and the
continued fraction expansion plays a central role in the estimation of
the discrepancy of n-alpha-sequences, which link the last task with
the first one and closes the circle of our considerations.