Dimension, Digits, and Diophantine Approximation
PhD defence, Thursday 8 October 2026, Maiken Balman Gravgaard
During her PhD studies, Maiken Balman Gravgaard researched metric Diophantine approximation along with missing digits sets.
Diophantine approximation is the study of how well real numbers can be estimated by fractions. Some numbers can be approximated very accurately with relatively simple fractions while others are much more difficult to get close to. Understanding these differences is a classic problem in number theory and dynamical systems.
A number can be written in many different ways. In addition to the classical digit-representation, we can write them in binary, base 3, or using other representations such as continued fractions or Lüroth expansion.
If you look at the real numbers written in base 3 but only allow 0 and 2 as digits, you get the famous Cantor set – an iconic example of a fractal.
A central question in this thesis is how “large” such special sets of numbers actually are. Although such sets often contain infinitely many numbers, they might still be very sparse or have complicated fractal structure. Therefore, their size is measured with mathematical tools developed especially for these types of sets, namely Hausdorff measure and Hausdorff dimension. The thesis develops new results on the sizes of both sets of numbers represented by a restricted set of digits and sets of numbers that can be approximated unusually well or badly by fractions.
The PhD study was completed at department of mathematics, Faculty of Natural Sciences, Aarhus University.
This summary was prepared by the PhD student.
Time: Thursday, 8 October 2026 at 15:00
Place: Building 1532, room 116, Lecture Theatre G1, Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C
Title of PhD thesis: Metric Diophantine Approximation and Missing-Digits Sets
Contact information: Maiken Balman Gravgaard, e-mail: maiken@balmangravgaard.dk, tel.: +45 21524994
Members of the assessment committee:
Lecturer Demi Allen, Department of Mathematics and Statistics, University of Exeter, United Kingdom
Maitre de conférence Antoine Marnat, Laboratoire d'Analyse et de Mathématiques Appliquées, Université Paris Est Créteil, France
Professor Peter Jørgensen (chair), Department of Mathematics, Aarhus University, Denmark
Main supervisor: Professor Simon Kristensen, Department of Mathematics, Aarhus University, Denmark
Language: The PhD dissertation will be defended in English
The defence is public.
The PhD thesis is available for reading at the Graduate School of Natural Sciences/GSNS, Ny Munkegade 120, building 1521, 8000 Aarhus C