Improved bounds for the dimensions of planar distance sets

@article{Shmerkin2018ImprovedBF,
  title={Improved bounds for the dimensions of planar distance sets},
  author={Pablo Shmerkin},
  journal={Journal of Fractal Geometry},
  year={2018}
}
  • P. Shmerkin
  • Published 8 November 2018
  • Mathematics
  • Journal of Fractal Geometry
We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if $A$ has Hausdorff dimension $>1$, then the set of distances spanned by points of $A$ has Hausdorff dimension at least $40/57 > 0.7$ and there are many $y\in A$ such that the pinned distance set $\{ |x-y|:x\in A\}$ has Hausdorff… 

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