Areas spanned by point configurations in the plane

2020/08/31 by McDonald, Alex · 1 citation
#28A75 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2008.13720

Abstract

We consider an over-determined Falconer type problem on (k+1)-point configurations in the plane using the group action framework introduced in \citeGroupAction. We define the area type of a (k+1)-point configuration in the plane to be the vector in \R^\binomk+12 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⊂\R^d of sufficiently large Hausdorff dimension determines a positve measure set of area types.

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