Low-discrepancy sequences for piecewise smooth functions on the two-dimensional torus

TL;DR

This paper introduces explicit low-discrepancy sequences designed for efficiently approximating integrals of smooth, periodic functions over convex domains with positive curvature in two dimensions, leveraging diophantine approximation and Erdos-Turan inequality.

Contribution

It provides a novel construction of low-discrepancy sequences tailored for piecewise smooth functions on the 2D torus, with rigorous theoretical backing.

Findings

Sequences achieve low discrepancy for smooth functions on convex domains

The method relies on diophantine approximation techniques

The approach extends to a general version of the Erdos-Turan inequality

Abstract

We produce explicit low-discrepancy infinite sequences which can be used to approximate the integral of a smooth periodic function restricted to a convex domain with positive curvature in R^2. The proof depends on simultaneous diophantine approximation and a general version of the Erdos-Turan inequality.