S. Hofmann, A. Iosevich
Falconer Conjecture in the Plane for Random Metrics
Preprint series: ESI preprints

MSC:

42B99 None of the above but in this section

Abstract: The Falconer conjecture says that if a compact planar
set has Hausdorff dimension $>1$, then the Euclidean distance set
$\Delta(E)=\{|x-y|: x,y \in E\}$ has positive Lebesgue measure. In
this paper we prove, under the same assumptions, that for almost
every ellipse $K$, $\Delta_K(E)=\{{||x-y||}_K: x,y \in E\}$ has
positive Lebesgue measure, where ${||\cdot||}_K$ is the norm
induced by an ellipse $K$. Equivalently, we prove that if a
compact planar set has Hausdorff dimension $>1$, then $\Delta(TE)$
has positive Lebesgue measure for almost every transformations $T$
with bounded positive eigenvalues. We also use this result to
deduce a version of the Erdos Distance Conjecture in the plane.

Keywords: Falconer conjecture, average decay, sectorial decomposition