On volume vectors determined by hypergraphs in thin subsets of Euclidean space

Abstract

Generalizing the Falconer distance problem, the authors of this paper recently established the first non-trivial dimensional threshold for any distance graph in high enough of a dimension. The methods developed were flexible enough to generalize from the Euclidean distance to any two point configuration, conditional on results on $k$-stars for the two point configuration. A natural question emerges on what happens to configurations that take in more than two points. In this paper we consider a classic three point variant of the Falconer distance problem, namely that on areas of triangles and its generalizations to volumes of simplices. In this model case we develop two methods. One we call the Jacobian method which allows us, through Heron's formula, to leverage earlier results on distance graphs and obtains non-trivial thresholds for volume vectors determined by a wide range of hypergraphs of simplices. Even in the classic case of the volume of a single simplex this method yields the best known dimensional thresholds if the dimension is considerably bigger than the size of the simplex. We develop a conjecture that has connections to rigidity theory. The Jacobian method works best in high dimensions so in the case of areas of triangles in the plane, we refine the work of Shmerkin and Yavicoli, who recently resolved a conjecture for areas of triangles in the plane, and obtain building blocks from which we can get abundance of area vectors determined by certain hypergraphs of triangles, such as chains of triangles connected on edges or vertices. The results improve and extend existing results of Galo and McDonald as well as of Greenleaf, Iosevich and Taylor.