# Hyperuniform point sets on flat tori: deterministic and probabilistic aspects

## TL;DR

This paper investigates hyperuniform point sets on flat tori, demonstrating their uniform distribution and analyzing various deterministic and probabilistic constructions, including QMC-designs and determinantal processes.

## Contribution

It establishes hyperuniformity for different classes of point sets on flat tori, linking it to uniform distribution and analyzing various construction methods.

## Key findings

- Hyperuniformity implies uniform distribution on flat tori.
- QMC-designs and certain probabilistic point sets are hyperuniform.
- Determinantal point processes exhibit hyperuniformity on flat tori.

## Abstract

In this paper we study hyperuniformity on flat tori. Hyperuniform point sets on the unit sphere have been studied by J.~Brauchart, P.~Grabner, W.~Kusner and J.~Ziefle. It is shown that point sets which are hyperuniform for large balls, small balls or balls of threshold order on the flat tori are uniformly distributed. Moreover, it is also shown that QMC--designs sequences for Sobolev classes, probabilistic point sets (with respect to jittered samplings) and some determinantal point process are hyperuniform.