On the Hausdorff dimension of pinned distance sets

@article{Shmerkin2017OnTH,
  title={On the Hausdorff dimension of pinned distance sets},
  author={Pablo Shmerkin},
  journal={Israel Journal of Mathematics},
  year={2017},
  volume={230},
  pages={949-972}
}
  • P. Shmerkin
  • Published 1 June 2017
  • Mathematics
  • Israel Journal of Mathematics
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 ∈ A} has full Hausdorff dimension for all x outside of a set of Hausdorff dimension 1 (in particular, for many x ∈ 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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