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arxiv: 2606.05482 · v1 · pith:V4ONFIRYnew · submitted 2026-06-03 · 🧮 math.NA · cs.NA

Infinite sequences with optimal diaphony, periodic L₂-discrepancy, and beyond

Peter Kritzer , Nicolas Nagel , Friedrich Pillichshammer This is my paper

Pith reviewed 2026-06-28 04:43 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords digital sequencesperiodic L2-discrepancydiaphonyquasi-Monte CarloBesov spacesinterlacingorder-2 sequencesextensible sequences
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The pith

Infinite order-2 digital sequences over F2 achieve the optimal order of periodic L2-discrepancy for all N greater than 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that infinite order-2 digital sequences over the field with two elements attain the lowest possible order of periodic L2-discrepancy, bounded above by a dimension-dependent constant times (log N) to the power d/2 divided by N. This upper bound holds for every N except 1 and matches known lower bounds for infinitely many such N. The construction improves earlier results that required order-5 sequences and reduces the dimension needed for the interlacing step from 5d to 2d. The same sequences also deliver explicit worst-case error rates when used for integration in periodic Besov spaces with dominating mixed smoothness.

Core claim

We prove that infinite order-2 digital sequences over F2 attain the optimal order L_{2,N}^{per}(S_d) ≤ C_d (log N)^{d/2}/N for all N in natural numbers except 1, matching known lower bounds for infinitely many N. This confirms the conjectured optimality of order-2 constructions. By this result we improve upon previously known constructions using order-5 digital sequences and reduce the underlying dimension for the interlacing construction from 5d to 2d. We establish our bounds within a broader framework of quasi-Monte Carlo integration for periodic Besov spaces S_{p,q}^r B(T^d) with dominating mixed smoothness r in (1/p,2), where the rules yield worst-case errors of order (log N)^{(d-1)(1-1/

What carries the argument

Order-2 digital sequences over F2 together with an interlacing construction applied in 2d dimensions.

If this is right

  • The sequences attain optimal diaphony as well as optimal periodic L2-discrepancy.
  • Interlacing constructions for higher-dimensional sequences can now start from 2d rather than 5d dimensions.
  • Worst-case integration errors in the periodic Besov spaces are bounded by (log N)^{(d-1)(1-1/q)} / N^{min(r,1)} when r is not 1 and by (log N)^{d(1-1/q)}/N when r equals 1.
  • The sequences remain extensible in the number of points N while achieving these rates for every N greater than 1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The lower-dimensional interlacing step may make high-dimensional periodic integration more practical by reducing the size of the base sequences that must be generated.
  • The same order-2 sequences could be tested for optimality under other discrepancy measures or in non-periodic function spaces that share similar smoothness assumptions.
  • Because the bounds hold for all N except 1, these sequences can be used in adaptive or sequential sampling schemes without losing the optimal rate at any step.

Load-bearing premise

The known lower bounds on periodic L2-discrepancy apply in exactly the same setting as the upper bounds derived here, and the interlacing construction preserves the order-2 property when the dimension is reduced from 5d to 2d.

What would settle it

A direct computation for some large N where a lower bound is known to be attained that shows the periodic L2-discrepancy of an explicit order-2 sequence exceeds C_d (log N)^{d/2}/N.

read the original abstract

We investigate the periodic $L_2$-discrepancy of infinite sequences $\S_d$ in $[0,1)^d$ and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over $\mathbb{F}_2$ attain the optimal order $L_{2,N}^{{\rm per}}(\S_d) \le C_d (\log N)^{d/2}/N$ for all $N \in \mathbb{N}\setminus \{1\}$, matching known lower bounds for infinitely many $N \in \mathbb{N}$. This confirms the conjectured optimality of order-2 constructions. By this result, we improve upon previously known constructions using order-5 digital sequences, and reduce the underlying dimension for the interlacing construction from $5d$ to $2d$, significantly improving practicality. We establish our bounds within a broader framework of quasi-Monte Carlo integration for periodic Besov spaces $S_{p,q}^rB(\mathbb{T}^d)$ with dominating mixed smoothness $r \in (1/p,2)$, where $p,q\in [1,\infty]$. Rules based on infinite order-2 digital sequences yield worst-case errors of order $(\log N)^{(d-1)(1-1/q)} / N^{ \min(r,1)}$ for $r \not=1$, and $(\log N)^{d(1-1/q)}/N$ for $r=1$, for all $N \in \mathbb{N}\setminus\{1\}$, while preserving extensibility in $N$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that infinite order-2 digital sequences over F_2 attain the optimal order L_{2,N}^{per}(S_d) ≤ C_d (log N)^{d/2}/N for all N ∈ N {1}, matching known lower bounds for infinitely many N. This confirms the conjectured optimality of order-2 constructions. The result is embedded in a framework for quasi-Monte Carlo integration over periodic Besov spaces S_{p,q}^r B(T^d) with dominating mixed smoothness r ∈ (1/p,2), yielding worst-case errors of order (log N)^{(d-1)(1-1/q)} / N^{min(r,1)} (r ≠ 1) and (log N)^{d(1-1/q)}/N (r=1) for all N > 1 while preserving extensibility; the interlacing dimension is reduced from 5d to 2d.

Significance. If the central proof holds, the result is significant: it supplies the first explicit infinite constructions achieving the conjectured optimal order for periodic L_2-discrepancy (upper bound for every N>1, matching lower bounds infinitely often) and simultaneously improves the practical dimension of interlacing while extending the error analysis to a range of periodic Besov spaces. The combination of matching rates, extensibility, and dimension reduction strengthens the case for order-2 digital sequences in high-dimensional QMC.

minor comments (2)
  1. The transition between the discrepancy result and the Besov-space error bounds (abstract, final paragraph) would benefit from an explicit statement of how the periodic L_2-discrepancy controls the worst-case error in S_{p,q}^r B.
  2. Notation for the sequence S_d and the constant C_d is introduced in the abstract without a forward reference to the precise definition in the main text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its significance for optimal periodic L2-discrepancy constructions and dimension reduction in interlacing, and recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives an upper bound on periodic L2-discrepancy for infinite order-2 digital sequences over F_2 that matches known external lower bounds for infinitely many N, within a framework of quasi-Monte Carlo integration on periodic Besov spaces. The derivation is a direct mathematical proof establishing the stated order for all N>1 while preserving extensibility; no equations reduce by construction to fitted inputs, no parameters are renamed as predictions, and no load-bearing uniqueness theorems or ansatzes are imported via self-citation chains. The result is self-contained against the cited lower bounds and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the result rests on standard properties of digital sequences and known lower bounds in discrepancy theory; no free parameters, invented entities, or ad-hoc axioms are mentioned.

axioms (1)
  • domain assumption Standard properties of infinite order-2 digital sequences over F_2 and known lower bounds on periodic L2-discrepancy
    The proof relies on these as background for the upper bound construction.

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