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 F├╝hrer (Austria; since July 2021)
    Mentors: Wolfgang Woess, Stefan Lendl.

  • Benjamin Klahn (Denmark; January 2020–February 2023)
    Mentors: Alfred Geroldinger, Marc Technau.
  • 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.

  • Second phase of the doctoral program:
  • Stefan Planitzer (Austria; July 2015–August 2018)
    Mentors: Robert Tichy, Wolfgang Woess, 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).

Associated Students

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

  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.