Digital almost nets

TL;DR

This paper introduces smaller 'almost nets' that nearly match digital nets in distribution properties but are exponentially more compact, and provides bounds on their minimal size.

Contribution

It constructs almost nets that are significantly smaller than digital nets while maintaining similar distribution properties and establishes lower bounds on their size.

Findings

Almost nets contain nearly the expected points in dyadic boxes.

Constructed almost nets are exponentially smaller than digital nets.

A lower bound on the size of almost nets is proven.

Abstract

Digital nets (in base $2$) are the subsets of $[0,1]^d$ that contain the expected number of points in every not-too-small dyadic box. We construct sets that contain almost the expected number of points in every such box, but which are exponentially smaller than the digital nets. We also establish a lower bound on the size of such almost nets.