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The star discrepancy of a union of randomly digitally shifted Korobov polynomial lattice point sets depends polynomially on the dimension

Josef Dick, Friedrich Pillichshammer

Abstract

The star discrepancy is a quantitative measure of the uniformity of a point set in the unit cube. A central quantity of interest is the inverse of the star discrepancy, $N(\varepsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most~$\varepsilon$ in dimension~$s$. It is known that $N(\varepsilon, s)$ depends only linearly on the dimension~$s$. All known proofs of this result are non-constructive. Finding explicit point set constructions that achieve this optimal linear dependence on the dimension remains a major open problem. In this paper, we make progress on this question by analyzing point sets constructed from a multiset union of digitally shifted Korobov polynomial lattice point sets. Specifically, we show the following two results. A union of randomly generated Korobov polynomial lattice point sets shifted by a random digital shift of depth $m$ can achieve a star discrepancy whose inverse depends only linearly on $s$. The second result shows that a union of all Korobov polynomial lattice point sets, each shifted by a different random digital shift, achieves the same star discrepancy bound. While our proof relies on a concentration result (Bennett's inequality) and is therefore non-constructive, it significantly reduces the search space for such point sets from a continuum of possibilities to a finite set of candidates, marking a step towards a fully explicit construction.

The star discrepancy of a union of randomly digitally shifted Korobov polynomial lattice point sets depends polynomially on the dimension

Abstract

The star discrepancy is a quantitative measure of the uniformity of a point set in the unit cube. A central quantity of interest is the inverse of the star discrepancy, , defined as the minimum number of points required to achieve a star discrepancy of at most~ in dimension~. It is known that depends only linearly on the dimension~. All known proofs of this result are non-constructive. Finding explicit point set constructions that achieve this optimal linear dependence on the dimension remains a major open problem. In this paper, we make progress on this question by analyzing point sets constructed from a multiset union of digitally shifted Korobov polynomial lattice point sets. Specifically, we show the following two results. A union of randomly generated Korobov polynomial lattice point sets shifted by a random digital shift of depth can achieve a star discrepancy whose inverse depends only linearly on . The second result shows that a union of all Korobov polynomial lattice point sets, each shifted by a different random digital shift, achieves the same star discrepancy bound. While our proof relies on a concentration result (Bennett's inequality) and is therefore non-constructive, it significantly reduces the search space for such point sets from a continuum of possibilities to a finite set of candidates, marking a step towards a fully explicit construction.

Paper Structure

This paper contains 14 sections, 9 theorems, 71 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Notation
  3. Korobov polynomial lattice point set.
  4. Digital shifts.
  5. Walsh functions.
  6. Main results
  7. Proofs
  8. Auxilliary results
  9. Star-discrepancy estimate
  10. The proof of Theorem \ref{['thm1']}
  11. Local discrepancy variance estimate.
  12. Applying Bernstein's inequality.
  13. Existence result.
  14. The proof of Theorem \ref{['thm2']}

Key Result

Theorem 1.2

Let $m\in\mathbb{N}$ and let $p\in\mathbb{Z}_2[x]$ be an irreducible polynomial of degree $m$. Let $q_1,\dots,q_{2^m}\in G_m$ and $\boldsymbol{\sigma}_1,\dots,\boldsymbol{\sigma}_{2^m}\in \mathbb{Q}_{2^m}^s$ be independent and uniformly distributed. Define the multiset and write $N:=|P|=2^{2m}$. Then, for every $\delta\in(0,1)$, with probability at least $\delta$, the star discrepancy of $P\subse

Theorems & Definitions (16)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 6 more