I have just uploaded a paper on the Hausdorff dimension of pinned distance sets. For a given set , the distance set of E is
If , the pinned distance set of E with respect to x is
An important problem in geometric measure theory is to understand the size of and, more generally, . Falconer conjectured that, if , then . Even in the plane, this is still open (although there has been a recent surge in progress on this question).
In this post we focus on the planar case. Liu showed that, if
then for “most” (formally defined later) points in the plane,
Shortly thereafter, Shmerkin improved Liu’s result when is close to 1. Specifically, he showed that if the dimension of E is strictly greater than 1, then
for most points .
In this post, I will describe the main argument in my paper on pinned distance sets, where I proved the following.
Theorem 1: If is analytic and has Hausdorff dimension strictly greater than one, then
for all x outside a set of Hausdorff dimension at most one.
Update: It is possible to generalize this bound so that it depends on the Hausdorff dimension of . In particular, we can conclude that
The proof of this generalization is essentially the same as the proof of this post, with one or two technical additions. In this post I will sketch the proof of Theorem 1.Continue reading