### Project 13: "Additive and combinatorial number theory"

*This project is running since 2015.*

Principal investigator: Christian Elsholtz

Graz University of Technology, Austria.

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

### DK Student

**Second phase of the doctoral program:**-
**Stefan Planitzer**(Austria; since July 2015)

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

*Mentors*: Robert Tichy, Daniel Krenn.

*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).

### 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.