Projection theorems using effective dimension

In this post I will describe how to use effective dimension to investigate how the fractal dimension of a set changes when projected onto a linear subspace. The first theorem investigating this behavior is the celebrated Marstrand projection theorem.

Theorem (Marstrand’s projection theorem): Let E \subseteq \mathbb{R}^2 be an analytic set. Then for almost every angle \theta,

\text{dim}_H(p_\theta E) = \min\{1, \text{dim}_H(E)\},

where

p_\theta(x, y) = x\cos \theta + y\sin\theta.

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The dimension spectra of planar lines

I recently uploaded a paper solving the dimension spectrum conjecture. I wanted to write a post explaining some of the intuition behind the proof.

The theme of the conjecture is to better understand the randomness of points on a line. Let (a, b) be a slope-intercept pair. The dimension spectrum of a line (a,b) is set of all effective dimensions of points on that line, i.e.,

\text{sp}(L_{a,b}) = \{\text{dim}(x, ax + b) \, \vert \, x \in [0, 1]\}.

In the early 2000s, Jack Lutz conjectured that the dimension spectrum of any line in the plane contains a unit interval.

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Effective dimension of points on a line

The purpose of this post is to describe the strategy Neil Lutz and I used to prove lower bounds on the effective dimension of points on a line in \mathbb{R}^2. This strategy seems quite general, and has been the basis for a variety of results (e.g. https://arxiv.org/pdf/2102.00134.pdf, https://arxiv.org/pdf/1711.02124.pdf and https://arxiv.org/pdf/1701.04108.pdf). 

In this post I will focus on the effective dimension of points. However, one of primary motivations of this work is that  lower bounds on effective dimension can be converted into lower bounds on the (classical) Hausdorff dimension of sets. For example, Neil and I used the lower bounds on the effective dimension of points on a line to give better bounds on certain classes of Furstenberg sets. In a future post I might describe how to convert effective theorems into classical theorems using the point-to-set principle of Jack and Neil Lutz.

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Hello world

Welcome to my Math + CS blog! I plan on using this blog to talk about my research and to post lecture notes.

Quick test of Latex functionality:

dim_H = \min\limits_{A \subseteq \mathbb{N}} \sup\limits_{x\in E} dim^A(x).

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