Pinned Distance Sets Using Effective Dimension

TL;DR

This paper applies algorithmic and complexity tools to analyze the fractal dimension of pinned distance sets, establishing new lower bounds for sets with Hausdorff dimension greater than one.

Contribution

It introduces a novel approach using effective dimension and Kolmogorov complexity to improve bounds on the Hausdorff dimension of pinned distance sets.

Findings

Pinned distance sets have Hausdorff dimension at least 3/4 for most points outside a small exceptional set.

Improves previous bounds for sets with Hausdorff dimension close to one.

Applicable to analytic sets in the plane with dimension greater than one.

Abstract

In this paper, we use algorithmic tools, effective dimension and Kolmogorov complexity, to study the fractal dimension of distance sets. We show that, for any analytic set $E\subseteq\R^2$ of Hausdorff dimension strictly greater than one, the \textit{pinned distance set} of $E$, $\Delta_x E$, has Hausdorff dimension of at least $\frac{3}{4}$, for all points $x$ outside a set of Hausdorff dimension at most one. This improves the best known bounds when the dimension of $E$ is close to one.