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on Falconer distance problem in $\mathbb{R}^2$ ($C_{34}$)

The statement

The Falconer distance problem threshold $C_{34} = s_{\Delta}(\mathbb{R}^2)$ in the plane is defined as

s_{\Delta}(\mathbb{R}^2) :\ :=\ \inf\Bigl\{\, s\in[0,2]\ :\ \forall\ \text{compact }E\subset\mathbb{R}^2,\ \dim_H(E)>s\ \Longrightarrow\ \lvert\Delta(E)\rvert>0 \,\Bigr\}.

where for a compact set $E\subset \mathbb{R}^2$ , the distance set is

\Delta(E)\ :=\ \{\,\lvert x-y\rvert\ :\ x,y\in E\,\}\ \subset\ [0,\infty)

[GIOW2018-def-distance-set], $\dim_H$ denotes Hausdorff dimension, and $\lvert\Delta(E)\rvert$ denotes the 1-dimensional Lebesgue measure of $\Delta(E)$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.