# Band-limited mimicry of point processes by point processes supported on a lattice

## TL;DR

This paper investigates when a point process on the real line can be approximated by a lattice-supported point process at a given bandwidth, providing complete solutions for Poisson processes and partial results for the sine process, with implications for number theory.

## Contribution

It characterizes the parameter ranges where point processes can be mimicked by lattice-supported processes at a specific bandwidth, including complete results for Poisson and partial results for sine processes.

## Key findings

- Complete characterization for Poisson processes.
- Existence and nonexistence regions for sine process.
- Applications to zeros of the Riemann zeta function.

## Abstract

We say that one point process on the line $\mathbb{R}$ mimics another at a bandwidth $B$ if for each $n \ge 1$ the two point processes have $n$-level correlation functions that agree when integrated against all bandlimited test functions on bandwidth $[-B, B]$. This paper asks the question of for what values $a$ and $B$ can a given point process on the real line be mimicked at bandwidth $B$ by a point process supported on the lattice $a\mathbb{Z}$. For Poisson point processes we give a complete answer for allowed parameter ranges $(a,B)$, and for the sine process we give existence and nonexistence regions for parameter ranges. The results for the sine process have an application to the Alternative Hypothesis regarding the scaled spacing of zeros of the Riemann zeta function, given in a companion paper.