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

This project is running since 2015.

Principal investigator: Christian Elsholtz
Graz University of Technology, Austria.
Mentor for: Wiegel, Technau; Iaco, Kreso, Smertnig.

### Project description (pdf-file)

Christian Elsholtz is particularly interested in the interplay of additive and multiplicative questions in number theory. The methods he uses are often a combination of tools from combinatorics and analytic number theory. In this area Elsholtz proved, for example, that the set of primes (or other multiplicatively defined sets, such as smooth numbers) cannot be written as a ternary sumset A+B+C, even if finitely many deviations are allowed. In this connection he has made a methodological contribution to the theory of sieves. As a very different example of interplay between additive and multiplicative questions Elsholtz studied with Buttkewitz the values of multiplicative functions (such as the Liouville function) along arithmetic progressions, and proved with Dietmann new upper bounds on the maximal size of Hilbert cubes in the set of squares (and other sets), by studying the connection between iterated sumsets and arithmetic progressions. Of his particular interest are Diophantine equations, often studied from analytic point of view. He is further interested in zero sum problems and, quite generally, in extremal sets with (or without) certain patterns.