### Project 09: "Diophantine approximation and combinatorial problems"

*This project is running since 2010.*

Principal investigator: Robert Tichy

Graz University of Technology, Austria.

*Mentor for*: Planitzer; Bazarova, Raseta, Smertnig.

Associated senior scientist: István Berkes (since 2015)

Graz University of Technology, Austria.

*Mentor for*: Candellero, Iaco, Temmel.

Associated scientist: Sophie Frisch (since 2010)

Graz University of Technology, Austria.

Associated scientist: Christoph Aistleitner (since 2018)

Graz University of Technology, Austria.

*Mentor for*: Ddamulira.

### DK Students

**Second phase of the doctoral program:****Niclas Technau**(Germany; May 2015 - May 2018)

*Email*: technau@math.tugraz.at

*Mentors*: Christian Elsholtz, Mario Weitzer.**First phase of the doctoral program:**-
**Fabrizio Barroero**(Italy; October 2010–November 2013)

Personal homepage;*Email*: barroero@math.tugraz.at

*Mentors*: Alfred Geroldinger, Clemens Heuberger.

*PhD Defense*: November 22, 2013.

*Referees*: C. Fuchs (Salzburg), R. Tichy, J. Vaaler (Austin).

*Examiners*: C. Fuchs (Salzburg), R. Tichy.

*Thesis Title*: "Diophantine approximation: analytic and geometric methods".

*PhD Defense*: June 25, 2018.

*Referees*: C. Elsholtz, R. Tichy, G. Larcher (Linz), V. Berasnevich (York).

*Examiners*: R. Tichy, G. Larcher (Linz), V. Berasnevich (York).

### Associated Students

**Second phase of the doctoral program:****Mahadi Ddamulira**(Uganda; since November 2017)

*Email*: mddamulira@tugraz.at

*Mentors*: Christian Elsholtz, Christoph Aistleitner

*Thesis Title*: "Diophantine equations and linearly recurrent sequences".**Kwok Chi Chim**(Hongkong; September 2014 - August 2018)

*Email*: chim@math.tugraz.at

*Mentors*: Gisbert Wüstholz, Christopher Frei.

*Thesis Title*: "Linear forms in logarithms and applications to Diophantine problems".

*PhD Defense*: April 23, 2018.

*Referees*: C. Fuchs (Salzburg), G. Wüstholz (Zurich), R. Tichy.

*Examiners*: C. Fuchs (Salzburg), G. Wüstholz (Zurich).**Adrian Scheerer**(Germany; January 2014–April 2017)

*Email*: scheerer@math.tugraz.at

*Mentors*: Jörg Thuswaldner, Manfred Madritsch.**First phase of the doctoral program:**-
**Christopher Frei**(Austria; October 2009–October 2011)

Personal homepage;*Email*: frei@math.tugraz.at

*Mentors:*Clemens Heuberger, Johannes Wallner.

*PhD Defense*: October 11, 2011.

*Referees*: G. Lettl (KFU Graz), R. Tichy.

*Examiners*: G. Lettl (KFU Graz), R. Tichy. -
**Maria Rita Iacò**(Italy; October 2012–December 2014)

*Email*: iaco@mat.unical.it

*Mentors*: Istvan Berkes, Christian Elsholtz.

*PhD Defense*: December 10, 2014.

*Referees*: L. Colzani (Milano), M. Drmota (Vienna)

*Examiners*: I. Carbone (Calabria), L. Colzani (Milano), M. Drmota (Vienna), R. Tichy. -
**Dijana Kreso**(Croatia; October 2010–March 2014)

Personal homepage;*Email*: kreso@math.tugraz.at

*Mentors:*Christian Elsholtz, Alfred Geroldinger

*PhD Defense*: March 21, 2014.

*Referees*: A. Dujella (Zagreb), C. Fuchs (Salzburg), R. Tichy.

*Examiners*: A. Dujella (Zagreb), R. Tichy.

*Thesis Title*: "Dynamical systems and normal numbers: qualitative and computational aspects".

*PhD Defense*: April 24, 2017.

*Referees*: S. Thonhauser (TU Graz), P. Grabner (TU Graz), J. Thuswaldner (MU Leoben).

*Examiners*: P. Grabner (TU Graz), J. Thuswaldner (MU Leoben).

### Project description

The central research topic of this project is diophantine approximation
with applications to algorithmic number theory and discrepancy theory. Methods from various fields of mathematics
are used, in particular from harmonic and complex analysis, from approximation theory as well
as from algebra, combinatorics and probability theory. The group of R. F. Tichy works on
the solution of Diophantine equations, where he intensively cooperates with various coauthors,
e.g. Y. Bilu (Bordeaux), G. Wüstholz (ETH Zurich), C. Fuchs (Univ. Salzburg) and A. Petho
(Debrecen). Recently they worked on applications to computational geometry, jointly with M.
Kerber and M. Weitzer. Tichy organized a special workshop at the Erwin Schrödinger Institute
Vienna "Normal Numbers: Arithmetic, Computational and Probabilistic Aspects" (with V.
Becher, Buenos Aires; Y. Bugeaud, Strasbourg and T. Slaman, Berkeley), which will be
extended to a semester program at IMS Singapore in spring 2019.
Jointly with M. Drmota (TU Vienna) R.F. Tichy published the monograph "Sequences,
Discrepancies, and Applications" (in the Springer Lecture Notes in Mathematics series, vol.
1651, 1997) on discrepancy theory and uniform distribution modulo 1. The research interests
of R.F. Tichy also include questions from the theory of automatic sequences, dynamical
systems and corresponding digital expansions and arithmetic functions: we mention cooperations
with J.-P. Allouche (Paris), V. Bergelson (Columbus, Ohio), J. Rivat and C. Mauduit
(Marseille), R. Tijdeman (Leiden). During the last years a useful combination of methods
from Diophantine approximation and from probability theory was developed. This yields new
insight to old problems from metric discrepancy theory and from the analysis of multivariate
numerical integration. In this area there is an intensive cooperation with C. Aistleitner (TU
Graz), I. Berkes (TU Graz and Renyi Institute Budapest) and with various national and international
research groups. Furthermore, methods from Diophantine analysis could be applied
to algebraic problems concerning units in number fields, to lattice point problems, and to
the analysis of pseudo-random numbers. This is even useful for specific problems in financial
mathematics.

Showcases for possible PhD themes can be found here.