Dimensional lower bounds for Falconer type incidence theorems

Let 1 ≤ k ≤ d and consider a subset E ⊂ ℝ^d. In this paper, we study the problem of how large the Hausdorff dimension of E must be in order for the set of distinct noncongruent k-simplices in E (that is, noncongruent point configurations of k + 1 points from E) to have positive Lebesgue measure. This generalizes the k = 1 case, the well-known Falconer distance problem and a major open problem in geometric measure theory. Many results on Falconer type theorems have been established through incidence theorems, which generally establish sufficient but not necessary conditions for the point configuration theorems. We establish a dimensional lower threshold of d+12 on incidence theorems for k-simplices where k ≤ d ≤ 2k + 1 by generalizing an example of Mattila. We also prove a dimensional lower threshold of d+12 on incidence theorems for triangles in a convex setting in every dimension greater than 3. This last result generalizes work by Iosevich and Senger on distances that was built on a construction by Valtr. The final result utilizes number-theoretic machinery to estimate the number of solutions to a Diophantine equation.