Random lattice vectors in a set of size O(n)

TL;DR

This paper investigates the statistical properties of vectors in high-dimensional random lattices within sets of volume proportional to the dimension, using sieve techniques and exploring applications to number theory.

Contribution

It introduces a novel approach combining sieve methods to analyze lattice vectors in high dimensions and applies these findings to number theory problems.

Findings

Derived statistical properties of lattice vectors in high dimensions

Established connections between lattice vector distributions and number theory

Provided new bounds and probabilistic estimates for lattice vectors

Abstract

We adopt the sieve ideas of Schmidt and S\"odergren in order to study the statistics of vectors of a random lattice of dimension n contained in a set of volume O(n). We also give some sporadic applications of our results to number theory.