Pinned Distance Sets Using Effective Dimension

D. M. Stull

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arXiv · 2207.12501

Pinned Distance Sets Using Effective Dimension

D. M. Stull

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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.

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D. M. Stull. 2022-07-25. Pinned Distance Sets Using Effective Dimension. https://arxiv.org/abs/2207.12501

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