BME Fractal Geometry Seminar

Past Seminars

December 2025

Monday, December 1, 2025

10:05–10:55 am, room 306 in Building H, BME.

Effective Methods in Fractal Geometry, Part II

Ryan Bushling (BME)

Abstract

In this second lecture, we will give two applications of the point-to-set principle with the goal of seeing the varieties of proof technique that arise and to acquire a sense of what types of results effective methods are suited for proving. The first application will pertain to intersections of sets, and the second will involve constructing a set with extreme projection properties. Both applications will depend crucially on symmetry of information—undoubtedly the second most important tool in effective methods after the point-to-set principle itself.

November 2025

Monday, November 17, 2025

10:05–10:55 am, room 306 in Building H, BME.

Effective Methods in Fractal Geometry, Part I

Ryan Bushling (BME)

Abstract Board YouTubeNotes by Ryan

Effective methods in geometric measure theory and fractal geometry originated in a 2018 paper of J. Lutz and N. Lutz on the Kakeya conjecture. Since then, mathematicians and theoretical computer scientists have applied these tools to prove new distance set bounds, partially resolve the Furstenberg set conjecture, establish sharp radial projection estimates, and much more. During the course of these achievements, this algorithmic approach has developed and matured into a theory with characteristic tricks, techniques, and lemmas.

This is the first in a series of \(N \geq 2\) talks on effective methods with the goal of laying down the foundations for a general fractal geometry audience. In this lecture, we will begin with the fundamentals: oracle machines, Kolmogorov complexity, the point-to-set principles, and symmetry of information. These basic objects and results are all we will need to prove a simple pinned distance set theorem for arbitrary norms, generalizing a result of Falconer.

Monday, November 3, 2025

10:15–10:55 am, room 306 in Building H, BME.

Multi-scale properties of infinitely generated self-similar attractors

Alex Rutar (University of Jyväskylä)

Abstract Board YouTube

The box dimension of a (finitely generated) self-similar set always exists. However, perhaps surprisingly, this is not the case in the infinitely generated case. If we only consider the lower and upper box dimensions in isolation, the situation seems mysterious. But if we take into account information about all scales simultaneously, a cohesive picture emerges. One way to understand this phenomenon is to use a certain Lipschitz function called the branching function which one can associate with general subsets of Euclidean space. This function is ubiquitous in (continuous) incidence geometry but its appearance in the context of dynamically invariant sets is quite surprising. Based on joint work with A. Banaji.

October 2025

Monday, October 20, 2025

10:15–10:55 am, room 306 in Building H, BME.

Topology of univoque sets in double-base expansions

Vilmos Komornik (University of Strasbourg)

Abstract Slides

This is a joint work with Yuru Zou and Yichang Li. Given two real numbers \(q_0, q_1>1\) satisfying \(q_0+q_1 \geq q_0 q_1\) and two real numbers \(d_0 \neq d_1\), by a double-base expansion of a real number \(x\) we mean a sequence \(\left(i_k\right) \in\{0,1\}^{\infty}\) such that $$ x=\sum_{k=1}^{\infty} \frac{d_{i_k}}{q_{i_1} q_{i_2} \cdots q_{i_k}} . $$ We denote by \(U_{q_0, q_1}\) the set of numbers \(x\) having a unique expansion. Its topological properties have been investigated in the equal-base case \(q_0=q_1\) for a long time. We extend this research to the case \(q_0 \neq q_1\). While many results remain valid, a great number of new phenomena appear due to the increased complexity of double-base expansions.