Star Discrepancy Bounds of Double Infinite Matrices induced by Lacunary Systems

TL;DR

This paper extends star discrepancy bounds to double infinite matrices generated by lacunary sequences, showing they achieve similar low discrepancy with fewer digits needed for simulation, compared to previous random matrix models.

Contribution

It introduces a new class of matrices based on lacunary sequences that maintain low star discrepancy bounds with reduced computational complexity.

Findings

Achieves star discrepancy bounds comparable to random matrices.

Reduces the number of digits needed for simulation.

Provides probabilistic guarantees for the discrepancy bounds.

Abstract

In 2001 Heinrich, Novak, Wasilkowski and Wo\'zniakowski proved that the inverse of the star discrepancy satisfies $n(d,\varepsilon)\leq c_{\abs}d \varepsilon^{-2}$ by showing that there exists a set of points in $[0,1)^d$ whose star-discrepancy is bounded by $c_{\abs}\sqrt{d/N}$. This result was generalized by Aistleitner who showed that there exists a double infinite random matrix with elements in $[0,1)$ which partly are coordinates of elements of a Halton sequence and partly independent uniformly distributed random variables such that any $N\times d$-dimensional projection defines a set $\{x_1,\ldots,x_N\}\subset [0,1)^d$ with \begin{equation*} D^*_N(x_1,\ldots,x_N)\leq c_{\abs}\sqrt{d/N}. \end{equation*} In this paper we consider a similar double infinite matrix where the elements instead of independent random variables are taken from a certain multivariate lacunary sequence and prove that with high probability each projection defines a set of points which has up to some constant the same upper bound on its star-discrepancy but only needs a significantly lower number of digits to simulate.