### 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:

1. 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)
2. 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)
3. Constructing large sets of integers such that no a_i divides a sum a_j+a_k, (arxiv:1609.07935)
4. Regularity of distribution of prime numbers (arxiv:1602.04317, arxiv:1702.00289)
5. Maximum order of iterated multiplicative functions (arxiv:1709.04799)
6. Subsetsums which are square-free or similar, (see for example arxiv:1510.05260, arxiv:1601.04754)
Future work can be related to the topics above, but can also include a variety of topics in additive combinatorics or combinatorial number theory, for example:
• 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.