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 $E\subseteq\R^2$ 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.