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.

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