# Verification report — OWR-17469-002 (Novak: the asymptotic constant for integration on C^k(D_d))

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete answer** to the question as posed in the source (item (3) of Novak's
abstract in Oberwolfach Report 6/2020, under the report's standing hypothesis that D_d is a bounded Lipschitz domain
of volume 1); every theorem, proposition and lemma is proved (the remarks marked as expectations, in Remarks
7.2-7.4, are not), and the scope is as follows.
- **Settled.** For all k, d >= 1 there is c_{k,d} in (0, infinity), defined on the unit cube by the derivatives of
  order k alone, such that lim_n n^{k/d} e_n(C^k(D), INT) = c_{k,d} lambda(D)^{1+k/d} for every bounded open set D
  with the cone condition, in particular for every bounded Lipschitz domain (Theorem 1.1(c)). So the limsup is a
  limit, and at volume 1 it is the same number for all such D. The lower-order bounds |D^a f| <= 1, |a| < k, do not
  change the constant.
- **Lower bound without regularity.** liminf_n n^{k/d} e_n(C^k(D), INT) >= c_{k,d} lambda(D)^{1+k/d} for every
  bounded open set D (Theorem 1.1(b)).
- **A boundary hypothesis is needed (d >= 2).** There are bounded, connected, Jordan measurable comb domains of volume
  1 with n^{k/d} e_n -> infinity (Theorem 1.1(d)).
- **Not determined.** The value of c_{k,d} and its dependence on d (items (1), (2) of the report); whether liminf and
  limsup can differ for a bounded open set without the cone condition.
- **Closest prior work.** Polovinkin (1975, 1989) proved the corresponding asymptotics for the Sobolev classes
  L^m_p, 1 < p < infinity, with a Euclidean-type seminorm; the case p = infinity with the full C^k norm is not covered
  there. No priority is claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** E. Novak, "Optimal algorithms for some high dimensional problems" (joint work with A. Hinrichs,
  D. Krieg, J. Prochno, M. Ullrich, H. Woźniakowski), extended abstract in Oberwolfach Report 6/2020, *New
  Perspectives and Computational Challenges in High Dimensions*, Oberwolfach Rep. 17(1) (2020) 377–416 (published
  2021), doi:10.4171/OWR/2020/6; the abstract is on pp. 386–388. Read in the open PDF of the report.
  - p. 386: algorithms use n function values, and D_d is a bounded Lipschitz domain with volume 1 (standing
    hypothesis). The minimal error e_n(F, INT) and, as a known fact, its Bakhvalov–Smolyak representation for unit
    balls of norms (inf over nodes of sup of the integral over the functions vanishing at the nodes).
  - p. 387: C^k(D_d): all partial derivatives up to order k bounded by 1 (the tilde class, with all directional
    derivatives, is a different class). Item "(3) OP" asks whether limsup_n e_n(C^k(D_d), INT) n^{k/d} depends on
    D_d and whether it is always a limit; for k = 1 the report recalls
    lim_n e_n(C^1(D_d), INT) n^{1/d} = zeta_d lambda_d(D_d) [d! lambda_d(D_d)/2^d]^{1/d}, zeta_d -> 1.
- **Proposer's later texts.** E. Novak, J. Complexity 61 (2020) 101458 (arXiv 1908.05943), Remark 7(2): the guess
  that the limit exists and does not depend on D_d for Jordan measurable D_d of volume 1; Theorem 2 there (k = 1,
  Chernaya, Gruber) concerns the classes |f(x) − f(y)| <= ||x − y||_B for all x, y in D_d. E. Novak, MCQMC 2022
  proceedings (Springer 2024, pp. 105–131; arXiv 2307.06787), Remark 7(2): the same guess, now "possibly under some
  assumptions on the boundary".
- **Corpus record.** ulamai/UnsolvedMath, OWR-17469-002 (status `open`). Its clean statement asks whether the
  limsup depends on "the bounded domain D_d" and whether it is a limit (the source's Lipschitz hypothesis is not
  repeated there); its original statement also asks for the dependence on d, which is items (1)–(2) of the report
  and is not addressed.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Item (3) under the standing hypothesis (bounded Lipschitz domain, volume 1) | the limit exists and equals c_{k,d} for every such D | Theorem 1.1(a), (c) |
| Arbitrary volume | limit c_{k,d} lambda(D)^{1+k/d} (the form of the report's k = 1 formula) | Theorem 1.1(c) |
| "Bounded domain" read without any boundary hypothesis (the corpus wording) | lower bound for every bounded open set; for d >= 2 the limit can be +infinity (combs), so the answer then needs a boundary hypothesis | Theorem 1.1(b), (d) |
| Class read as C^k(D), C^k(closure D) or W^{k,inf}(D); nodes in D or in closure D | same asymptotic constant | Remark 1.2, end of Section 5 |
| Dependence on d (original statement; items (1)–(2) of the report) | not addressed | Section 7 |
| Connectedness of D | not used; the combs are connected | Theorem 1.1 |

## Results in the paper
- **Theorem 1.1(a)** (Propositions 3.1, 3.3; Lemma 3.2). On the unit cube, with E_n the minimal radius of the class
  S(Q) defined by the derivatives of order k: c_{k,d} = lim n^{k/d} E_n = inf_n n^{k/d} E_n in (0, infinity); the same
  limit for the Dirichlet class S_0 (functions of class C^k on R^d vanishing outside Q). Tools: Taylor's formula along
  segments (Lemma 2.2), unisolvence of the principal lattice (Lemma 2.3), a local estimate (Lemma 2.4),
  self-similarity and Zador's sandwich, and a cutoff ("firewall") with a Leibniz estimate (Lemma 3.2).
- **Theorem 1.1(b)** (Lemma 4.1, Proposition 4.2). Rescaled Dirichlet fooling functions glued over disjoint dyadic
  cubes in D belong to the full class (all derivatives up to order k bounded by 1); Jensen's inequality and
  exhaustion of D give the lower bound for every bounded open set.
- **Theorem 1.1(c)** (Lemmas 5.1, 5.2, Proposition 5.3). Under the cone condition the boundary is null, and a boundary
  layer of cells equipped with principal lattices, inside cones at every point, costs o(n) nodes and o(n^{-k/d}) in
  the error. The upper bound uses only the derivatives of order k.
- **Theorem 1.1(d)** (Lemma 6.1, Proposition 6.2). Comb domains D_2 x (0,1)^{d−2} with teeth of widths
  c_0 j^{−1−delta}, 0 < delta < k/d, rescaled to volume 1: e_n >= const (n+1)^{−delta}.
- **Remarks 7.1–7.4.** Explicit bounds (c_{1,1} <= 1/4); the case k = 1 (c_{1,d} at most the l^1 quantization
  coefficient; equality expected, not proved and not used); the case d = 1 (c_{k,1} is also the periodic constant;
  the value K_k (2 pi)^{−k} is stated only as an expectation); other seminorms (remark only); open questions.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The exact program of the note** (`check_note.py`, standard library, exact rationals, about 2 s; TOTAL failures:
  0). Unisolvence of the principal lattice (42 cases up to N_k = 126, with negative controls); the multinomial
  identity of Lemma 2.2; the counts and measure bounds of Lemma 3.2 and the union of its cells; the inequalities for
  M = ceil((2n)^{1/d}) in Proposition 3.1(e); the identity in the proof of Lemma 5.2; the comb tail bound and the
  normalisation of c_0 in Proposition 6.2; the value 1/(4n) in Remark 7.1.
- **Consistency computations of the second verification run** (`verification_run_B/`, numpy/scipy/sympy; all six
  programs re-run on 2026-10-20 with identical outputs). d = 1, k <= 4: LP values with equidistant nodes bracket
  K_k (2 pi)^{−k} (k = 2, n = 64: 0.031055 <= 1/32 <= 0.031470); locally optimised nodes give n^k E_n decreasing and
  above K_k (2 pi)^{−k}; the full class on (0, L) gives the same n^k r / L^{k+1} for L = 1/2, 1, 2, 4 once the
  lower-order bounds are inactive. d = 2, k = 1: l^1 geodesic distance integrals of diamond lattices (upper bounds for
  the radii) for the unit square and an L-shape of area 3/4 approach sqrt(2)/3 = 0.4714 after normalisation. d = 2,
  k = 2: a finite-difference LP gives n r / lambda^2 for square and L-shape differing by about 0.0075/m.

## Independent verification runs
The results were first obtained with complete analytic proofs. Two independent verification runs, both
AI-assisted, followed on 19–20 October 2026; each read the source and wrote its own programs. A third independent
verification run, also AI-assisted, examined the final text of the note on 20 October 2026 (programs in
reproducibility/independent_run_2/).

| Item | Run A | Run B | Run 3 (final text) |
|---|---|---|---|
| Statement, class, standing hypothesis, item (3) against the source | CONFIRMED (report PDF) | CONFIRMED (report PDF downloaded again; both later papers of the proposer read) | CONFIRMED (report, arXiv 1908.05943v2 and 2307.06787v1 fetched again and read) |
| Theorem 1.1(a) (cube constant, firewall) | CONFIRMED (re-derived line by line) | CONFIRMED (re-derived; d = 1 LP consistent) | CONFIRMED (re-derived from the text of the note) |
| Theorem 1.1(b) (lower bound, every bounded open set) | CONFIRMED | CONFIRMED | CONFIRMED |
| Theorem 1.1(c) (limit under the cone condition) | CONFIRMED | CONFIRMED (d = 1 and d = 2 computations consistent) | CONFIRMED (including the argument for W^{k,inf}) |
| Theorem 1.1(d) (combs, d >= 2) | CONFIRMED | CONFIRMED | CONFIRMED |
| Readings of the class and of the nodes | CONFIRMED (answer independent of the reading) | — | CONFIRMED |
| Finite ingredients | CONFIRMED by own exact computation (k, d <= 5) | CONFIRMED by own exact computation, with negative controls | CONFIRMED by own programs (exact and symbolic; k, d <= 5; d <= 6, M <= 60), with negative controls; d = 1 LPs consistent |
| Remark on k = 1 (identification with the l^1 quantization constant) | NOT CHECKED (approximation step not written) | consistent (d = 2 computation) | labelled correctly as not proved |
| Re-run of the package | — | — | all programs re-run from the source archive, identical outputs |
| Novelty | residual risk: Polovinkin's papers not accessible to this run | Polovinkin 1975 and 1989 read in the Russian originals: closest prior work, p < infinity only | Polovinkin read again (copies fetched at the writing; Math-Net.ru returned HTTP 504); further searches: nothing new |
| Overall | CONFIRMED with six editorial corrections | CONFIRMED with corrections of the literature paragraph and bibliography; no change to the theorem or the proof | CONFIRMED with corrections of citations and wording; no mathematical error |

Corrections required and made (details in the note's Verification paragraph): the cone condition in d = 1 is
stated for bounded open intervals; the readings of the class and of the nodes are stated (Remark 1.2, with an
approximation argument for W^{k,inf} at the end of Section 5); the k = 1 identification is labelled as neither proved
nor used; "d >= 2" is stated wherever the comb appears; "for n large" is explicit in the proof of Proposition 5.3;
the description of Polovinkin (1975, 1989) and the positioning of the result as their p = infinity counterpart; the
bibliographic data (all DOIs checked with Crossref); the remark that the classes of Novak (2020), Theorem 2, are
defined by Lipschitz conditions between all pairs of points.

Corrections required by the third run and made: Novak (2020), Theorem 2, rests on Chernaya, East J. Approx. 1 (1995)
47–60 (the paper Novak cites), which is now cited besides the related Ukr. Mat. Zh. paper; the existence of the limit
along all n is no longer listed as a difference from Polovinkin (1989), whose Theorem 2 holds for all N; the
relation between the Lipschitz classes of Novak (2020) and the classes of C^1 functions is stated precisely; the
sentence on scope in this report and the description of the free radius for n < N_k in the reproducibility README
were made precise.

## Relation to the literature, novelty and scope
- **Closest prior work.** V. I. Polovinkin, Sib. Math. J. 16(6) (1975) 963–968 (Russian original Sib. Mat. Zh.
  16:6 (1975) 1255–1262), doi:10.1007/BF00967394: for L^m_p(Omega), 1 < p < infinity, p > max(n/m, 1), Omega bounded
  with piecewise smooth boundary, the seminorm (int (sum_{|a|=m} m!/a! |D^a f|^2)^{p/2})^{1/p} and error
  functionals vanishing on polynomials of degree < m, the minimal error-functional norm is
  N^{−m/n} (mes Omega)^{1/q + m/n} B (1 + o(1)) along some sequence N -> infinity. V. I. Polovinkin, Sib. Math. J.
  30(2) (1989) 289–297 (Russian 151–160), doi:10.1007/BF00971385, Theorem 2: for all N, with a weight g in L_q, pm > n,
  domains including all domains star-shaped with respect to a ball: N^{−m/n} T ||g||_{L_r(Omega)} (1 + o(1)),
  r = qn/(qm + n); the constant-weight case is attributed to the 1975 paper and to a 1978 Tashkent note (not read).
  Both papers were read in the Russian originals on Math-Net.ru (open full texts) at the writing and by the second
  verification run, and again by the third run (in the copies fetched at the writing, which are identical to those
  of the second run; Math-Net.ru returned HTTP 504 at the time of the third run). What the note adds for
  p = infinity: p = infinity is excluded there and the passage p -> infinity is not automatic; the class is the full
  C^k norm (maximum of sup norms of all partial derivatives up to order k), and the lower-order bounds are shown not
  to matter; the limit holds for every bounded open set with the cone condition (all bounded Lipschitz domains) and
  the lower bound for every bounded open set (the boundary conditions of the 1989 paper are those of Polovinkin 1983,
  not read); the combs show that a boundary hypothesis is needed for C^k(D). Like the 1989 paper (and unlike the
  1975 paper) the note has the limit along all n.
- **k = 1.** The dependence on the volume is known (the report's formula; Novak 2020, Theorem 2, from Chernaya,
  East J. Approx. 1 (1995) 47–60, and Gruber, Adv. Math. 186 (2004); a related paper: Chernaya, Ukr. Math. J. 47
  (1995), Hölder classes on Jordan measurable sets). These results concern Lipschitz-type classes
  |f(x) − f(y)| <= ||x − y||_B for all x, y in D: for the l^1 norm and convex D the C^1 functions with |d_i f| <= 1
  belong to such a class and are uniformly dense in it (up to |f| <= 1); for non-convex D they need not belong to it.
  Weighted first-order results: Babenko–Babenko–Kovalenko, Res. Math. 29(2) (2021) 15–27, and Kovalenko, Mat. Stud.
  61(1) (2024) 84–96.
- **Method.** The quantization scheme of Zador (1982) and Graf–Luschgy (LNM 1730, 2000), adapted to higher
  smoothness; unisolvence of the principal lattice (Chung–Yao 1977, Nicolaides 1972); Lipschitz implies the cone
  condition (Adams–Fournier, 4.6 and 4.11; Grisvard, Thm 1.2.2.2), cited as standard and not re-read.
- **Searches (19–20 October 2026).** arXiv API queries (asymptotic constants for integration, cubature, sampling;
  asymptotically optimal cubature; Zador and integration; exact asymptotics; cubature on Lipschitz and general domains;
  Kovalenko, Babenko); works citing Novak 2020 and the 2023 survey in Semantic Scholar and OpenAlex (titles and
  abstracts checked); zbMATH reviews of Chornaya, Derets', Boikov, Polovinkin 1988, Ramazanov, Reztsov, Osipov, Yumova,
  Noskov; Crossref; a few web searches. Third run (20 October): works citing Novak 2020 (6) and the 2023 survey (5)
  in OpenAlex; seven further arXiv API queries; zbMATH title searches (asymptotically optimal cubature, 1990–2026;
  Borodachov) and the reviews of both Chernaya papers of 1995 (Zbl 0848.41023, 0886.41023); all 16 DOIs again with
  Crossref; one web search, which found only texts that list the question as open. No printed answer to item (3)
  for k >= 2 was found in the sources searched; a search that finds nothing is not a proof of novelty.
- **Not read.** Polovinkin 1978 (Tashkent note) and 1983; Ramazanov, Sov. Math. Dokl. 15 (1974) 789–793 (grid
  formulas); Sobolev–Vaskevich, The Theory of Cubature Formulas (Kluwer 1997); Babenko 1976–77; Gruber 2004 (cited
  through Novak 2020); the two Chernaya papers of 1995 beyond their zbMATH reviews.
- **Scope.** The note answers item (3) under the report's hypothesis. The value of c_{k,d}, its dependence on d, and
  the behaviour without the cone condition are not determined. No priority is claimed.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/