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Erratum to "Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands"

This erratum corrects specific proof steps and statements in a previous paper regarding higher-order scrambled digital nets, confirming the validity of the principal convergence rate result for smooth integrands while withdrawing a flawed theorem on finite-difference variation and providing corrected variance bounds and logarithmic factors.

Original authors: Josef Dick

Published 2026-08-26

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CC BY 4.0

Original authors: Josef Dick

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of high-dimensional mathematics, scientists often face a problem that feels like trying to measure the volume of a shape with hundreds of invisible walls. They need to calculate the total value of a function that changes in complex ways across many different directions at once. This is a task known as numerical integration, and it is essential for everything from predicting weather patterns to pricing financial assets. The difficulty lies in the sheer number of points required to get an accurate answer; as the number of directions increases, the amount of work needed to get a precise result can explode. To solve this, mathematicians use special sets of points, arranged with a specific kind of order, to sample the function. These are called digital nets. To make these nets even better, researchers apply a technique called scrambling, which shuffles the points in a controlled way to smooth out errors, much like how a baker might fold dough to ensure ingredients are evenly distributed. The goal is always the same: to get the most accurate answer possible with the fewest number of samples.

A significant paper published in 2011 by mathematician Josef Dick claimed to have found the ultimate solution for a specific class of these problems. The paper argued that a particular method, using what are known as order-d nested-uniformly scrambled digital nets, could achieve the fastest possible rate of error reduction for smooth functions. This result was celebrated because it promised a near-perfect efficiency for high-dimensional calculations. However, a new note from the same author, published in 2026, serves as a formal correction to that earlier work. It does not overturn the main success story, but it does withdraw several specific claims and proof steps that were found to be flawed. The core finding remains solid: the method works and achieves the optimal speed of convergence. But the path to proving it required a complete reworking of the underlying logic, and some of the tools originally used to measure the smoothness of the functions were discarded entirely.

The original paper had relied on a specific way of measuring the "roughness" or variation of a function, using a concept called finite-difference variation. The author now admits that this measurement did not actually match the mathematical definition of the smoothness norm it was supposed to represent. In simpler terms, the ruler used to measure the function's complexity was not the same as the standard ruler accepted by the field. Because of this mismatch, the proof that relied on this specific variation could not hold up. The author explicitly withdraws the theorem that was based on this variation. Furthermore, a specific step in the proof involving how the scrambled points interact with each other contained a missing square in a variance bound, and the logic used to determine the power of a logarithmic factor in the error rate was insufficient. These were not minor typos but fundamental gaps in the argument that required a new approach.

To fix these issues, the author has replaced the flawed sections with a direct proof based on the unanchored mixed Sobolev norm. This is a standard and well-understood way of measuring how smooth a function is, focusing on its mixed partial derivatives. By building the argument directly on this established norm, the proof avoids the pitfalls of the previous variation method. The new proof confirms that for functions with square-integrable mixed partial derivatives up to a certain order, the scrambled digital nets still achieve the optimal rate of error reduction. The error decreases at a speed determined by the minimum of the function's smoothness and the order of the scrambling, multiplied by a logarithmic factor. This confirms that the method is indeed as powerful as originally hoped, but the mathematical justification is now cleaner and more robust.

The correction also clarifies how the points are generated and how their randomness is handled. The original text had described a process involving an inverse map that was not well-defined for all points. The new note replaces this with a clear, step-by-step definition of how the points are scrambled and interlaced, ensuring that the resulting set of points is uniformly distributed without needing any impossible mathematical inverses. It also corrects the way the covariance, or the relationship between different points in the set, is calculated. These adjustments ensure that the statistical properties of the point set are exactly as described, removing any ambiguity about how the randomness is applied.

Ultimately, this erratum is a story of scientific integrity and precision. It shows that even when a major result is correct, the path to proving it can contain errors that must be acknowledged and fixed. The main conclusion—that these scrambled digital nets provide the best possible performance for smooth integrands—stands firm. The numerical experiments and the optimal algebraic exponent for the error rate remain unchanged. The only things that have changed are the tools used to prove it and the specific details of the mathematical machinery. By withdrawing the incorrect claims about the finite-difference variation and providing a direct, corrected proof, the author has ensured that the foundation of this important result is solid. For researchers relying on these methods, the takeaway is clear: the method works, the error rates are optimal, and the mathematical reasoning behind it has been rigorously repaired.

Technical Summary of the Erratum

Problem Statement
This note addresses critical errors in the original paper "Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands" (J. Dick, Annals of Statistics 39, 2011). The original work claimed that order- $d$ nested-uniformly scrambled digital $(t, m, s)$ -nets achieve a root mean square error (RMSE) of $O(N^{-\min(\alpha, d) - 1/2 + \varepsilon})$ for functions in unanchored Sobolev spaces with mixed partial derivatives up to order $\alpha$ . While the principal result regarding the optimal algebraic rate remains valid, the original paper contained fundamental flaws in its proof machinery. Specifically, the claimed equivalence between a proposed finite-difference variation and the Sobolev norm was incorrect, the covariance structure used in the variance decomposition relied on an ill-defined inverse interlacing map, and the variance bound in Theorem 10 contained a missing square and an incorrect logarithmic factor for the case $d > \alpha$ .

Methodology and Corrections
The erratum replaces the flawed arguments with rigorous corrections across several key areas:

Scrambling and Interlacing Definitions: The original definition of order- $d$ scrambling involved an inverse interlacing map $D_d^{-1}$ , which is not well-defined as $D_d$ is not surjective. The correction defines the order- $d$ scrambled point $Y$ directly as $Y = D_d(\Pi(z))$ , where $\Pi$ is Owen's nested uniform scrambling applied to $ds$ coordinates, and $D_d$ is the digit interlacing map. Proposition 2.1 establishes that $Y$ is uniformly distributed on $[0, 1)^s$ without requiring an inverse map.
Covariance Structure: A new covariance statement (Proposition 2.2) is derived for the nested uniform scrambling. This proposition calculates the expectation of products of Walsh functions under scrambling using a block-indexed function $\chi_\ell$ . This replaces the previous indexing in Lemma 6 and handles identical points without ad-hoc conventions.
Variance Decomposition: The variance of the estimator is re-derived using the corrected covariance. The original Equation (3.2) incorrectly summed over all frequencies; the corrected version (Equation 10) sums over non-zero frequencies. Lemma 3.1 provides the corrected variance decomposition:
$\text{Var}[\hat{I}_N(f)] = \sum_{\ell \in \mathbb{N}_0^{ds} \setminus \{0\}} \sigma_{d,\ell,s}^2(f) \Gamma_{d,\ell}(P)$
where $\sigma_{d,\ell,s}^2(f)$ represents the block energy of the Walsh coefficients and $\Gamma_{d,\ell}(P)$ is the gain coefficient dependent on the net structure.
Withdrawal of Finite-Difference Variation: The paper explicitly withdraws the claim that the finite-difference variation $V_\alpha$ coincides with the displayed Sobolev norm. A counterexample ( $f(x) = x^2/2$ ) demonstrates that the variation (a supremum over partitions) does not equal the integral of the squared derivative. Furthermore, the admissibility of increments in the original Appendix B proof was shown to be flawed. Consequently, all results relying on $V_\alpha$ are withdrawn.
Direct Sobolev Estimate: To replace the withdrawn finite-difference argument, a direct Sobolev estimate (Lemma 5.2) is provided. This lemma bounds the block energy $\sigma_{d,\ell,s}(f)$ directly in terms of the unanchored mixed Sobolev norm $\|f\|_{H^{s,\alpha}}$ . The proof utilizes a finite Walsh-block identity, resolving parent cells into fine children and applying Minkowski's integral inequality to bound the finite differences by the mixed partial derivatives.
Corrected Logarithmic Factor: The proof for the case $d > \alpha$ is revised to correctly account for the decay of the Walsh blocks. The original theorem stated an incorrect logarithmic power. The corrected Theorem 6.2 establishes the variance bound:
$\text{Var}[\hat{I}_N(f)] \le C_{b,s,\alpha,d} \|f\|_{H^{s,\alpha}}^2 b^{-(2\rho+1)M} (M+2)^{ds-1}$
where $\rho = \min(\alpha, d)$ and $M = m-t$ .

Key Results
The primary result of the erratum is the validation of the optimal convergence rate for the corrected proof.

RMSE Rate: For a family of nets with fixed base $b$ , dimension $s$ , order $d$ , and smoothness $\alpha$ , the root mean square error satisfies:
$\text{RMSE} = O\left(N^{-\rho - 1/2} (\log N)^{(ds-1)/2}\right) = O\left(N^{-\rho - 1/2 + \varepsilon}\right)$
for every $\varepsilon > 0$ , where $\rho = \min(\alpha, d)$ .
Optimality: The algebraic exponent $-\rho - 1/2$ remains optimal for the mixed Sobolev class.
Specific Corrections:
- The variance bound in Theorem 10 is corrected to include a square on the Sobolev norm.
- The logarithmic factor is corrected from the original statement to $(ds-1)/2$.
- Typographical and indexing errors regarding the digit interlacing map, the domain of $E_d$ , and the definition of finite differences are rectified in Section 7.

Significance and Claims
The author states that the principal smooth-function result of the original paper remains valid: order- $d$ nested-uniformly scrambled digital nets achieve the optimal rate of convergence for smooth integrands. The numerical experiments and the optimal algebraic exponent cited in the original work are unchanged. However, the specific proof path involving the finite-difference variation is entirely discarded as invalid. The significance of this erratum lies in providing a mathematically sound, direct Sobolev-based proof that restores the theoretical foundation of the convergence rates, correcting the variance bounds and logarithmic factors without altering the fundamental conclusion regarding the efficiency of higher-order scrambled digital nets.

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