[Paper Review] $L_p$-discrepancy and beyond of higher order digital sequences

Josef Dick, Aicke Hinrichs+2 more|arXiv (Cornell University)|Jan 27, 2016

Mathematical Approximation and Integration17 references3 citations

TL;DR

This paper establishes the optimality of $L_p$-discrepancy for order 2 digital $(t,d)$-sequences over the binary field, proving they achieve the best possible order of magnitude for all $p \in (1,\infty)$, and further demonstrate that their discrepancy norms are optimal in bounded mean oscillation, exponential Orlicz, Besov, and Triebel-Lizorkin spaces.

ABSTRACT

The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution modulo one of infinite sequences. In 1986 Proinov proved for all $p>1$ a lower bound for the $L_p$-discrepancy of general infinite sequences in the $d$-dimensional unit cube, but it remained an open question whether this lower bound is best possible in the order of magnitude until recently. In 2014 Dick and Pillichshammer gave a first construction of an infinite sequence whose order of $L_2$-discrepancy matches the lower bound of Proinov. Here we give a complete solution to this problem for all finite $p > 1$. We consider so-called order $2$ digital $(t,d)$-sequences over the finite field with two elements and show that such sequences achieve the optimal order of $L_p$-discrepancy simultaneously for all $p \in (1,\infty)$. Beyond this result, we estimate the norm of the discrepancy function of those sequences also in the space of bounded mean oscillation, exponential Orlicz spaces, Besov and Triebel-Lizorkin spaces and give some corresponding lower bounds which show that the obtained upper bounds are optimal in the order of magnitude.

Motivation & Objective

• To close the gap in understanding whether Proinov's $L_p$-discrepancy lower bound is tight in order of magnitude for all $p > 1$.
• To construct sequences achieving the optimal $L_p$-discrepancy order for all $p \in (1,\infty)$, resolving a long-standing open problem.
• To extend the analysis of discrepancy norms beyond $L_p$ spaces to include bounded mean oscillation, exponential Orlicz, Besov, and Triebel-Lizorkin spaces.
• To provide matching lower bounds in these function spaces, confirming the optimality of the upper bounds obtained.

Proposed method

• Construction of order 2 digital $(t,d)$-sequences over the finite field $\mathbb{F}_2$ to ensure low discrepancy in high-dimensional uniform distribution.
• Use of advanced tools from harmonic analysis and function space theory to estimate the norm of the discrepancy function in $L_p$, BMO, Orlicz, Besov, and Triebel-Lizorkin spaces.
• Application of the theory of digital nets and sequences to derive explicit bounds on discrepancy in terms of the sequence parameters and dimension.
• Establishment of sharp upper bounds on discrepancy norms by leveraging the structure of the sequence and properties of the underlying function spaces.
• Derivation of matching lower bounds in each function space to confirm that the upper bounds are optimal in order of magnitude.
• Use of duality and interpolation techniques to relate discrepancy behavior across different function spaces and confirm consistency of optimality.

Experimental results

Research questions

• RQ1 Is Proinov's $L_p$-discrepancy lower bound for general sequences tight in order of magnitude for all $p > 1$?
• RQ2 Can a single sequence construction achieve the optimal $L_p$-discrepancy order simultaneously for all $p \in (1,\infty)$?
• RQ3 Are the discrepancy norms of such sequences optimal in function spaces beyond $L_p$, such as BMO and exponential Orlicz spaces?
• RQ4 Do the upper bounds on discrepancy norms in Besov and Triebel-Lizorkin spaces match known lower bounds in order of magnitude?
• RQ5 Can the optimality of the discrepancy behavior be established uniformly across multiple function space scales?

Key findings

• Order 2 digital $(t,d)$-sequences over $\mathbb{F}_2$ achieve the optimal order of $L_p$-discrepancy for all $p \in (1,\infty)$, resolving a long-standing open problem.
• The discrepancy function of these sequences has norms that are optimal in order of magnitude in the space of bounded mean oscillation (BMO).
• Upper bounds on the discrepancy norm in exponential Orlicz spaces are shown to be sharp, matching known lower bounds.
• The upper bounds on discrepancy norms in Besov and Triebel-Lizorkin spaces are optimal in order of magnitude, as confirmed by matching lower bounds.
• The construction simultaneously achieves optimal discrepancy behavior across all $L_p$ spaces and multiple function space scales, demonstrating robustness and universality.