Mean Value for Random Ideal Lattices

TL;DR

This paper compares the average number of lattice points in a ball for random ideal lattices in cyclotomic fields to that of general lattices, using advanced number theory tools, and demonstrates the existence of dense ideal lattice packings asymptotically.

Contribution

It establishes that ideal lattices in cyclotomic fields have similar point distributions to general lattices and proves the existence of dense ideal lattice packings asymptotically.

Findings

Average lattice point count matches that of general lattices.

Existence of ideal lattice packings with density approaching a specific asymptotic form.

Application of Hecke formula and Dedekind zeta bounds to lattice point analysis.

Abstract

We investigate the average number of lattice points within a ball for the $n$th cyclotomic number field, where the lattice is chosen at random from the set of unit determinant ideal lattices of the field. We show that this average is nearly identical to the average number of lattice points in a ball among all unit determinant random lattices of the same dimension. To establish this result, we apply the Hecke integration formula and subconvexity bounds on Dedekind zeta functions of cyclotomic fields. The symmetries arising from the roots of unity in an ideal lattice allow us to prove the existence of ideal lattice packings of dimension $\varphi(n)$ and density $n\cdot 2^{-\varphi(n)}(1+o(1))$ as $n$ goes to infinity.