Counting Co-Cyclic Lattices

TL;DR

This paper counts the number of co-cyclic lattices within integer lattices, showing they are prevalent with about 85% density, and connects this to complexity theory and lattice problem reductions.

Contribution

It provides an asymptotic count of co-cyclic lattices and establishes their dominant density among all lattices, extending previous lattice enumeration results.

Findings

Co-cyclic lattices constitute approximately 85% of all lattices.

The paper derives an explicit asymptotic formula for counting co-cyclic lattices.

Results have implications for complexity theory and lattice problem reductions.

Abstract

There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group $\mathbb{Z}^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq \mathbb{Z}^n$ such that $\mathbb{Z}^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $\left(\zeta(6) \prod_{k=4}^n \zeta(k)\right)^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.