### Project 13: "Additive combinatorics, number theory, extremal combinatorics"

*This project is running since 2015.*

Principal investigator: Christian Elsholtz

Graz University of Technology, Austria.

*Mentor for*: Bashir, Fadinger; Ddamulira, Wiegel, Technau; Iaco, Kreso, Smertnig.

### DK Students

**Third phase of the doctoral program:****Jakob Führer**(Austria; since July 2021)

*Email*: jakob.fuehrer@tugraz.at

*Mentors*: Wolfgang Woess, Stefan Lendl.**Benjamin Klahn**(Denmark; January 2020–February 2023)

*Email*: klahn@math.tugraz.at

*Mentors*: Alfred Geroldinger, Marc Technau.**Second phase of the doctoral program:**-
**Stefan Planitzer**(Austria; July 2015–August 2018)

*Email*: stefan.planitzer@student.tugraz.at

*Mentors*: Robert Tichy, Wolfgang Woess, Daniel Krenn.

*Thesis Title*: " Intersective polynomials, Diversity in rationally parameterised fields, and Probabilistic Galois theory".

*PhD Defense*: February 15, 2023.

*Chair*: Christian Elsholtz

*Referees*: Rainer Dietmann-Sopp (Royal Holloway, Univ of London), Joachim König (Korea National Univ. of Education).

*Examiners*: Rainer Dietmann-Sopp, Joachim König.

*Thesis Title*: "Sums of unit fractions, Romanov type problems and sequences with property P.".

*PhD Defense*: August 27, 2018.

*Chair*: A. Geroldinger.

*Referees*: C. Elsholtz, J. Schlage-Puchta (Rostock), Greg Martin (Vancouver).

*Examiners*: C. Elsholtz, J. Schlage-Puchta (Rostock).

### Associated Students

**Third phase of the doctoral program:****Gabriel Lipnik**(Austria; January 2020–May 2023)

*Email*: lipnik@math.tugraz.at

*Website*: https://www.gabriellipnik.at/

*Mentors*: Robert Tichy, Daniel Krenn.

*Thesis Title*: "On Integer Partitions, Caps, Progression-Free Sets and q-Regular Sequences".

*PhD Defense*: May 15, 2023.

*Chair*: Robert Tichy

*Referees*: Christian Elsholtz, Stephan Wagner (Uppsala University)

*Examiners*: Christian Elsholtz, Stephan Wagner

### Project description

I am particularly interested in the interplay of additive and
multiplicative questions. The methods used are often a combination of
tools from combinatorics and analytic number theory.

Topics that were studied in this project, in collaboration with Stefan
Planitzer, Niclas Technau, Rainer Dietmann, Florian Luca, Jan-Christoph
Schlage-Puchta, Igor Shparlinski, Marc Technau and Robert Tichy include:

- Number of solutions of Diophantine equation, e.g. the equation 1=1/x_1+ ... + 1/x_k, in integers x_i, (see arxiv:1805.02945)
- Questions what happens if one adds prime numbers to a thin sequence, such as powers of 2? (On Romanov's constant and Romanov type problem)
- Constructing large sets of integers such that no a_i divides a sum a_j+a_k, (arxiv:1609.07935)
- Regularity of distribution of prime numbers (arxiv:1602.04317, arxiv:1702.00289)
- Maximum order of iterated multiplicative functions (arxiv:1709.04799)
- Subsetsums which are square-free or similar, (see for example arxiv:1510.05260, arxiv:1601.04754)

- Solutions of diophantine equations.
- Sets of integers or lattice points with (or without) specified patterns, such as arithmetic progressions, (e.g. Roth-type results, cap sets etc).
- Prime numbers, sieve methods.
- Size of sum and product sets.