Pinned Distance Sets Using Effective Dimension

07/25/2022

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by   D. M. Stull, et al.

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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⊆^2 of Hausdorff dimension strictly greater than one, the pinned distance set of E, Δ_x E, has Hausdorff dimension of at least 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.