Abstract
In the former part of this dissertation, we establish generalizations and analogues of some probabilistic results in the geometry of numbers. One of our results states the following: for any $n \in \Z_{\geq 3}$ and any Borel subset $A$ of $\mathbb{R}^n$ with finite and nonzero volume, the probability that the set of primitive points of a random unimodular full-rank lattice in $\mathbb{R}^n$ does not contain any $\mathbb{R}$-linearly independent subset of $A$ of cardinality $(n-2)$ is bounded from above by a constant multiple, which depends only on $n$, of $\left(\mathrm{vol}(A)\right)^{-1}.$ This generalizes a result that is jointly due to J.\,S.~Athreya and G.\,A.~Margulis. We also establish upper bounds for the variances of certain counting functions defined on the space of unimodular full-rank lattices in $\mathbb{R}^n$; we then use these bounds to obtain analogues of independent results of C.\,A.~Rogers and W.\,M.~Schmidt. In the latter part of this dissertation, we prove an analogue of Khintchine's Theorem in metric Diophantine approximation:~given any regular (a notion to be defined) and nonincreasing function $\psi: \R_{\geq 0} \to \R_{>0}$ and any subhomogeneous function (a notion to be defined) $f: \R^n \to \R$, we establish a necessary and sufficient condition for ensuring that a generic element $f\circ g$ in the $G$-orbit of $f$ is asymptotically $\psi$-approximable. We also obtain a sufficient condition for ensuring that a generic element $f\circ g$ in the $G$-orbit of $f$ is uniformly $\psi$-approximable. Here, $G$ can be any closed subgroup of $\ASL$ that satisfies certain axioms whose definitions are inspired by the works of C.\,L.~Siegel and C.\,A.~Rogers.