On a mean value formula for multiple sums over a lattice and its dual

TL;DR

This paper generalizes Rogers' mean value formula to unimodular lattices and their duals, showing that in high dimensions, the lengths of vectors in a random lattice and its dual behave like independent Poisson processes.

Contribution

It extends Rogers' mean value formula to a broader setting involving lattices and their duals, and analyzes the asymptotic behavior of vector lengths in high dimensions.

Findings

Lengths of vectors in a random lattice converge to a Poisson process

Lengths of vectors in the dual lattice also converge to a Poisson process

The two processes are asymptotically independent in high dimensions

Abstract

We prove a generalized version of Rogers' mean value formula in the space $X_n$ of unimodular lattices in $R^n$, which gives the mean value of a multiple sum over a lattice $L$ and its dual $L^*$. As an application, we prove that for $L$ random with respect to the SL$(n,R)$-invariant probability measure, in the limit of large dimension $n$, the volumes determined by the lengths of the non-zero vectors $\pm x$ in L on the one hand, and the non-zero vectors $\pm x'$ in $L^*$ on the other hand, converge weakly to two independent Poisson processes on the positive real line, both with intensity 1/2.