## Pinned distance sets using effective dimension

I have just uploaded a paper on the Hausdorff dimension of pinned distance sets. For a given set $E \subseteq \mathbb{R}^n$, the distance set of E is

$\Delta E = \{ \| x-y\| \mid x,y\in E\}.$

If $x \in \mathbb{R}^n$, the pinned distance set of E with respect to x is

$\Delta_x E = \{\|x-y\| \mid y \in E\}.$

An important problem in geometric measure theory is to understand the size of $\Delta E$ and, more generally, $\Delta_x E$. Falconer conjectured that, if $\text{dim}_H(E) > n/2$, then $\mu(\Delta E) > 0$. Even in the plane, this is still open (although there has been a recent surge in progress on this question).

In this post we focus on the planar case. Liu showed that, if

$\text{dim}_H(E) = s > 1$

then for “most” (formally defined later) points in the plane,

$\text{dim}_H(\Delta_x E) \geq \frac{4}{3}s - \frac{2}{3}$.

Shortly thereafter, Shmerkin improved Liu’s result when $s$ is close to 1. Specifically, he showed that if the dimension of E is strictly greater than 1, then

$\text{dim}_H(\Delta_x E) > 0.69$

for most points $x$.

In this post, I will describe the main argument in my paper on pinned distance sets, where I proved the following.

Theorem 1: If $E \subseteq \mathbb{R}^2$ is analytic and has Hausdorff dimension strictly greater than one, then

$\text{dim}_H(\Delta_x E) \geq \frac{3}{4}$

for all x outside a set of Hausdorff dimension at most one.

Update: It is possible to generalize this bound so that it depends on the Hausdorff dimension of $E$. In particular, we can conclude that

$\text{dim}_H(\Delta_x E) \geq \frac{\text{dim}_H(E)}{4} + \frac{1}{2}$

The proof of this generalization is essentially the same as the proof of this post, with one or two technical additions. In this post I will sketch the proof of Theorem 1.

## 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

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

## 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.

## 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.

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