On the distribution of lengths of short vectors in a random lattice

TL;DR

This paper estimates how the lengths of the k-th shortest vectors in a high-dimensional random lattice are distributed, extending previous results to cases where k grows with the dimension n, using sieve theory techniques.

Contribution

It introduces a novel approach based on sieve theory to analyze the distribution of vector lengths in high-dimensional lattices, allowing k to increase with n.

Findings

Distribution estimates for k-th shortest vectors in high-dimensional lattices

Extension of previous results to larger k values as n increases

Improved bounds on vector length distributions

Abstract

We use an idea from sieve theory to estimate the distribution of the lengths of $k$th shortest vectors in a random lattice of covolume 1 in dimension $n$. This is an improvement of the results of Rogers and S\"odergren in that it allows $k$ to increase with $n$.