  Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. 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. We
establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer
type theorems for $k$-simplices. This threshold is nontrivial in the range $d/2
\leq k \leq d$ and is obtained through counting simplices in a standard lattice
using results of the Gauss circle problem. 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 also establish a dimensional lower threshold of $\frac{d+1}{2}$ on
incidence theorems for $k$-simplices where $k\leq d \leq 2k+1$ by generalizing
an example of Mattila. Finally, we prove a dimensional lower threshold of
$\frac{d+1}{2}$ 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.

Classical Analysis and ODEs;Number Theory