Probabilistic Star Discrepancy Bounds for Lacunary Point Sets

TL;DR

This paper investigates probabilistic bounds on the star discrepancy of lacunary point sets generated by a specific sequence, providing bounds that depend on the probability and improving understanding of their distribution properties.

Contribution

It introduces a new bound on the star discrepancy of lacunary sequences, linking it to probability and providing a potential construction method for low-discrepancy point sets.

Findings

Star discrepancy of lacunary sequences is bounded by C√(d log d / N) with high probability.

The bound depends explicitly on the probability of the inequality holding.

Provides a probabilistic construction approach for low-discrepancy point sets.

Abstract

By a result of Heinrich, Novak, Wasilkowski and Wo\'zniakowski the inverse of the star discrepancy $n(d,\varepsilon)$ satisfies $n(d,\varepsilon)\leq c_{\abs}d\varepsilon^{-2}$. Equivalently for any $N$ and $d$ there exists a set of $N$ points in $[0,1)^d$ with star discrepacny bounded by $\sqrt{c_{\abs}\cdot d/N}$. They actually proved that a set of independent uniformly distributed random points satisfies this upper bound with positive probability. Although Aistleitner and Hofer later refined this result by proving a precise value of $c_{\abs}$ depending on the probability with which the inequality holds, so far there is no general construction for such a set of points known. In this paper we consider the sequence $(x_n)_{n\geq 1}=(\langle 2^{n-1}x_1\rangle)_{n\geq 1}$ for a uniformly distributed point $x_1\in [0,1)^d$ and prove that the star discrepancy is bounded by $C\sqrt{d\log_2d/N}$. The precise value of $C$ depends on the probability with which this upper bound holds.