π-rate's (b)log

A few days ago, I uploaded a joint article with Balazs Barany and Henna Koivusalo to arXiv. The article is called Dynamical covering sets in self-similar sets and I will briefly summarise the paper here.

TLDR: Imagine you are trying to cover the circle $\mathbb{S}^1$ with randomly centred balls $B(x_k,r_k)$, where $x_k$ is distributed uniformly on $\mathbb{S}^1$ and $r_k$ is a decreasing sequence. The Dvoretzky covering question¹ asks for conditions on $r_k$ for the whole circle to be covered almost surely. Similarly, one can ask for when the cover is of full measure, or what size the covering set is.

In this article, we study an analogous dynamical problem for symbolic balls (cylinders) in self-similar sets, where we pick our initial point according to Bernoulli measures. We obtain a single pressure formula that describes the entire dimension theory, which shows distinct regions to the (known) homogeneous case.

From 9–24 November 2024, I taught at the National Higher School of Mathematics (NHSM) in Algiers, Algeria as part of the International Mathematics Masters Programme (IMM)¹. Over the course of two weeks, I taught part of the “Advanced Probability” module, focusing on martingales and branching processes.

My article Minkowski weak embedding theorem¹ with Stathis Chrontsios-Garitsis just got accepted at HJM, so it’s high time that I write a short summary.

The two sentence version: The Assouad embedding theorem loosely states that any metric space with finite Assouad dimension can be embedded into $\mathbb{R}^n$ without too much distortion. The Minkowski (or box-counting) dimension is smaller than the Assouad dimension and we show that an analogue of the Assouad embedding theorem still holds if one replaces the assumption of finite Assouad dimension with the weaker assumption of finite Minkowski dimension.