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Pinned Distance Sets Using Effective Dimension (2207.12501v2)
Published 25 Jul 2022 in cs.CC and math.CA
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 $E\subseteq\R2$ 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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