Pinned Distance Sets Using Effective Dimension
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.
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