Star Discrepancy Bounds of Double Infinite Matrices induced by Lacunary Systems

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.