Pinned Distance Sets Using Effective Dimension

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 $E\subseteq {\mathbb{R}}^2$ 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.