# Fibonacci lattices have minimal dispersion on the two-dimensional torus

## TL;DR

This paper demonstrates that Fibonacci lattices on the two-dimensional torus achieve minimal dispersion, matching the known lower bound, and characterizes all optimal integration lattices, providing insights into their structure.

## Contribution

It proves Fibonacci lattices attain the minimal dispersion bound and characterizes all sets achieving this bound, advancing understanding of optimal point configurations.

## Key findings

- Fibonacci lattices have dispersion exactly 2/n when n is a Fibonacci number.
- Fibonacci lattices meet the known lower bound for dispersion.
- The paper characterizes all integration lattices that achieve minimal dispersion.

## Abstract

We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality $n\ge 2$ in this setting is $2/n$. We show that if $n$ is a Fibonacci number then the Fibonacci lattice has dispersion exactly $2/n$ meeting the lower bound. Moreover, we completely characterize integration lattices achieving the lower bound and provide insight into the structure of other optimal sets. We also treat related results in the nonperiodic setting.