### 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.

*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**Kwok Chi Chim**(Hongkong; September 2014 - August 2018)

*Email*: chim@math.tugraz.at

*Mentors*: Gisbert Wüstholz, Christopher Frei.

*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.

*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.