Areas spanned by point configurations in the plane

Abstract

We consider an over-determined Falconer type problem on ( k + 1 ) (k+1) -point configurations in the plane using the group action framework introduced in by Greenleaf, Iosevich, Liu, and Palsson in [Rev. Mat. Iberoam. 31 (2015), pp. 799–810]. We define the area type of a ( k + 1 ) (k+1) -point configuration in the plane to be the vector in R ( k + 1 2 ) \mathbb {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 2 k − 1 2k-1 dimensional, and prove that a compact set E ⊂ R d E\subset \mathbb {R}^d of sufficiently large Hausdorff dimension determines a positve measure set of area types.