Improved bounds for the dimensions of planar distance sets

Abstract

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 dimension at least $29/42$ and lower box-counting dimension at least $40/57$. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input.