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arXiv · 1706.00131

On the Hausdorff dimension of pinned distance sets

Pablo Shmerkin
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Abstract

We prove that if $A$ is a Borel set in the plane of equal Hausdorff and packing dimension $s>1$, then the set of pinned distances $\{ |x-y|:y\in A\}$ has full Hausdorff dimension for all $x$ outside of a set of Hausdorff dimension $1$ (in particular, for many $x\in A$). This verifies a strong variant of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint $s=1$.

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Pablo Shmerkin. 2018-01-26. On the Hausdorff dimension of pinned distance sets. https://doi.org/10.1007/s11856-019-1847-9

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