### Project 04: "Fractal analysis and combinatorics of digital expansions"

*This project is running since 2010.*

Principal investigator: Peter Grabner

Graz University of Technology, Austria.

*Mentor for*: Alese, Zhang; Krenn, Minervino, Raseta.

### DK Student

**Second phase of the doctoral program:****Damir Ferizovic**(Bosnia and Herzegovina; since January 2017)

*Email*: damir.ferizovic@tugraz.at

*Mentors*: Jörg Thuswaldner, Johannes Wallner, Mario Weitzer.

**First phase of the doctoral program:**-
**Florian Greinecker**(Austria; October 2010-April 2016)

*Email*: greinecker@math.tugraz.at

*Mentors*: Clemens Heuberger, Jörg Thuswaldner.

### Project description

P. Grabner has been interested in the analysis
and probability on fractals since the early 1990s. In particular, he is interested in the
application of generating function techniques to this subject, which gives a close relation to
the iteration theory of rational functions. Furthermore, the proposer has contributed to the
study of digital expansions and their probabilistic and ergodic properties with a focus on
distribution properties of arithmetic functions related to digital expansions (e.g. additive and
multiplicative functions related to redundant numeration systems with application to cryptography).
His further research interests cover the mathematical analysis of algorithms, the
study of transition probabilities on graphs, functional equations with applications to physics
[Derfel et al. 2012], and distribution properties of points on spheres with application to numerical
integration. Grabner is a member of the recently prolonged special research programme
"Quasi-Monte Carlo methods: Theory and Applications" (2014-2022).

Showcases for possible PhD themes can be found here.