Some results on random unimodular lattices

Abstract

Let $n \in \mathbb{Z}_{\geq 3}.$ Given any Borel subset $A$ of $\mathbb{R}^n$ with finite and nonzero measure, we prove that the probability that the set of primitive points of a random full-rank unimodular 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 (see \cite[Theorem 2.2]{Log}). We also generalize independent results of C. A. Rogers (see \cite[Theorem 6]{MeanRog}) and W. M. Schmidt (see \cite[Theorem 1]{Metrical}) about primitive lattice points of random lattices to the case of primitive tuples of rank less than $\frac{n}{2}.$ In addition to the work of the authors who were just mentioned, a crucial element of this present paper is the usage of a rearrangement inequality due to Brascamp\textendash Lieb\textendash Luttinger (see \cite[Theorem 3.4]{BLL}).