Areas spanned by point configurations in the plane

TL;DR

This paper investigates the geometric properties of point configurations in the plane, specifically the space of area types spanned by these configurations, and establishes conditions under which large sets determine a positive measure of such area types.

Contribution

It introduces a new framework for analyzing area types of point configurations using group actions and characterizes the dimension of the space of all possible area types.

Findings

The space of all area types is 2k-1 dimensional.

Large Hausdorff dimension sets determine positive measure sets of area types.

Abstract

We consider an over-determined Falconer type problem on $(k+1)$-point configurations in the plane using the group action framework introduced in \cite{GroupAction}. We define the area type of a $(k+1)$-point configuration in the plane to be the vector in $\R^{\binom{k+1}{2}}$ with entries given by the areas of parallelograms spanned by each pair of points in the configuration. We show that the space of all area types is $2k-1$ dimensional, and prove that a compact set $E\subset\R^d$ of sufficiently large Hausdorff dimension determines a positve measure set of area types.