# Verification report — OWR-723-005 (Novak: randomly shifted lattices versus independent points, the expected p-th power of the L_p discrepancy)

Verification date: 2026-10-14.

**Verdict.** The note gives a **negative answer** to the question as posed in the source. Every statement is
proved, except Theorem 1.4, which is computer-assisted with exact arithmetic, and Conjecture 1.5, which is only
tested.
- **Settled.** For even p the inequality E_{z,Δ} disc_p^p(M_n^{z,Δ}) ≤ E_λ disc_p^p(M_n) is false. It fails
  for n = 3, d = 1, p = 4 (19/1620 > 16/1620), and for n = 3 and n = 4 with p = 4 in every dimension d (closed
  formulas, Theorem 1.2). For every d and every even p ≥ 4 it fails for all n ≥ N(d,p), with N(d,p) explicit,
  and the ratio of the two expectations is at least n^{p/2−1}/K(d,p) (Theorem 1.3). For the thirteen pairs
  (p,d) = (4,1..4), (6,1..4), (8,1..3), (10,1..2) it fails for every n ≥ 3 (Theorem 1.4, computer-assisted; the
  pairs (4,1) and (6,1) also without a computer). It holds with equality for p = 2 and for n ≤ 2 (Theorem 1.1).
- **Known in substance.** The equality for p = 2 rests on the pairwise independence of the points jz + Δ mod 1,
  the principle of random Weyl sampling (Sugita–Takanobu 2000; Sugita 2002). The question and the expansion of
  Lemma 2.1 are Novak's.
- **Scope.** The answer concerns E disc_p^p, the quantity of the source (equivalently the average L_p discrepancy
  (E disc_p^p)^{1/p}). For E disc_p, which the wording of the corpus record also allows, nothing is proved, and
  all computed cases go the other way. Theorems 1.3 and 1.4 are statements for fixed d and p; nothing is claimed
  for n polynomial in d, the regime that motivates the source, where the computed ratios are close to 1.
- **Open.** Failure for all n ≥ 3, d ≥ 1, even p ≥ 4 (Conjecture 1.5; exact values in 2,354 distinct triples
  (n,d,p), all positive); an upper bound E_lat = O(1/n); the comparison of E disc_p; the regime n = poly(d).

The note is unrefereed.

## Statement checked
- **Primary source.** E. Novak (joint work with A. Hinrichs), "New bounds for the star discrepancy", extended
  abstract in Oberwolfach Report 13/2004, *Discrepancy Theory and Its Applications* (organised by B. Chazelle,
  W. Chen and A. Srivastav, March 7–13, 2004), Oberwolfach Rep. 1(1) (2004), pp. 696–699 of 673–722,
  doi:10.4171/OWR/2004/13.
  - Read completely in the open PDF of the publisher (content.ems.press; 445,532 bytes, sha256
    `2f02cd5834ed1a868d95a20be9d172ade3b35e76caf8c9ff8ab6a3adaf6f3d3e`), fetched when the results were first
    obtained, by both verification runs, and again when the note was written (2026-10-14).
  - p. 697: the p-discrepancy disc_p(M_n) as the L_p norm of x_1⋯x_d − (1/n)Σ 1_{[0,x)}(t_i); the expansion of
    disc_p^p for even p; the expectation E_λ for Lebesgue measure on [0,1]^{nd}, with the factor (α/(α+j))^d.
  - p. 698: the generalized lattices with shift M_n^{z,Δ} = {t_j = jz + Δ (mod 1): j = 0, …, n−1}, z, Δ ∈ [0,1]^d;
    E_{z,Δ} as the integral over [0,1]^{2d}; the question whether E_{z,Δ}(disc_p^p(M_n^{z,Δ})) ≤ E_λ(disc_p^p(M_n));
    on pp. 698–699, the remark that an answer would need the two-dimensional integrals of
    max((j_i z + Δ)_1^α) with distinct j_i, α ∈ {1, …, p+1} and k ∈ {0, …, p}.
  - p. 699: the last sentence asks for an upper bound like (1) (star discrepancy ≤ c(d/n)^{1/2}) for lattices; a
    different problem, not treated.
  - No quantifiers are written; the expansion and the question are stated "for even p". The note reads the
    question as "for all n, d ≥ 1 and all even p" and also describes each fixed (d, p).
- **Corpus record.** ulamai/UnsolvedMath, OWR-723-005 (dataset version 1.6.0; upstream status `open`;
  literature triage of 2026-08-21 naming the source and Sinescu, J. Comput. Appl. Math. 232 (2009) 240–251).
  Its statement asks whether the expected Lp discrepancy of the generalized lattice point sets M_n(z,Δ) is at most
  that of the corresponding random point set M_n. It names neither the p-th power nor the parity of p; this is
  discussed under "Readings". The page range in the record, 673–722, is that of the whole report.
- **Conventions** (none matters): half-open or closed boxes (null sets); coincident points (a null set of
  (z, Δ)); index range j = 1, …, n instead of j = 0, …, n−1 (replace Δ by {Δ + z}); reduction mod 1 in each
  coordinate.

## Readings
| Reading | Answer | Where |
|---|---|---|
| For all n, d ≥ 1 and all even p (the source's question) | no; one certified triple: n = 3, d = 1, p = 4 | Theorem 1.2 |
| For each fixed even p | p = 2: yes, with equality; every even p ≥ 4: no, in every dimension, for all n ≥ N(d,p) | Theorems 1.1, 1.3 |
| For each triple (n, d, p) | equality for p = 2 and for n ≤ 2; failure for p = 4 with n ∈ {3, 4} and every d, for every n ≥ 3 in the thirteen pairs of Theorem 1.4, and for n ≥ N(d,p) in general; all other triples with n ≥ 3, p ≥ 4 tested only | Theorems 1.1–1.4, Conjecture 1.5, Remark 7.1 |
| The record's words, read as the average L_p discrepancy (E disc_p^p)^{1/p} (the usual meaning in the literature on random points) | the same answer as for the source's question | Section 1.1 |
| The record's words read literally as E disc_p (no p-th power) | not settled; every computed case goes the other way (lattices smaller), by quadrature in d = 1 and Monte Carlo in d = 1, 2, 3 | Remark 7.2 |
| Korobov generators z = g/n | not the question (z is uniform in the cube); for n = 3, d = 1, p = 4 the degenerate g = 0 alone contributes 1/45 > 16/1620 | Remark 7.3 |
| Fixed d and p with n → ∞, versus n polynomial in d with d large | the negative answer is for fixed d and p; nothing is claimed for n = poly(d), where the computed ratios are close to 1 (p = 4, n = 12: 1.0115 at d = 10, 1.000004 at d = 20) | Section 1.3 |

## Results in the paper
- **Lemma 2.1** (Novak's expansion), **Lemma 2.2** (law of the lattice points: pairwise uniformity; product over
  the coordinates; invariance of φ_α(S) under S → S + c and S → cS; φ_α(S) = α/(α+|S|) for |S| ≤ 2),
  **Lemma 2.3** (gap formula: E_Δ max_{q∈P}{q+Δ}^α = Σ_i (1 − (1−g_i)^{α+1})/(α+1)), **Corollary 2.4** (formulas
  for E_iid and E_lat − E_iid), **Proposition 2.5** (E_iid by set partitions without singletons; n^p E_iid is a
  polynomial in n of degree ≤ p/2; the formula for p = 4).
- **Theorem 1.1.** E_lat(n,d,2) = E_iid(n,d,2) = (2^{−d} − 3^{−d})/n; E_lat = E_iid for n ≤ 2.
- **Proposition 4.1.** φ_α({0,1,2}) = (α² + 1 − 2^{−α})/((α+1)(α+2)), below α/(α+3) for α ≥ 2; φ_1({0,1,3}) = 1/4,
  φ_2({0,1,3}) = 517/1296; φ_1({0,1,2,3}) = 11/54, φ_1({0,1,2,4}) = 85/432. **Remark 4.2**: φ_1(S) = 1/4 for every
  three-element S (proved with B_2); a closed form of φ_2 for triples (verified for all 7021 triples with
  b ≤ 119, not proved, not used); φ_1 − 1/5 takes both signs on four-element sets.
- **Theorem 1.2.** E_iid(n,d,4) = n^{−3}[(2^{−d} − 3^{−d}) + 3(n−2)(3^{−d} − 2·4^{−d} + 5^{−d})];
  E_lat(3,d,4) − E_iid(3,d,4) = (8/9)2^{−d}[(2/5)^d − (19/48)^d] > 0; the analogous closed form for n = 4 is
  positive for every d; for d = 1: 19/1620 > 16/1620 and 139/17280 > 99/17280.
- **Theorem 5.1.** E_lat(n,d,p) ≥ 4/((p+1)^d(p+2)(p+3)(n−1)) for n ≥ 3 and even p ≥ 2 (the progression event
  z < 1/(n−1) or z > 1 − 1/(n−1), and a Jensen reduction to d = 1).
- **Proposition 5.2.** E_iid(n,d,p) ≤ T_p(2^{−d} − 3^{−d})n^{−p/2}, with T_p the number of partitions without
  singleton blocks (4, 41, 715, 17722 for p = 4, 6, 8, 10; T_p ≤ (p/2)^p).
- **Theorem 1.3.** For every d and even p ≥ 4, failure for all n ≥ N(d,p) (least N ≥ 3 with N^{p/2−1} ≥ K(d,p),
  K = T_p(p+2)(p+3)(p+1)^d(2^{−d} − 3^{−d})/4); E_lat/E_iid ≥ n^{p/2−1}/K; no constant-factor version for fixed d,
  p. Values of N(d,p): 35, 146, 462 (p = 4) and 30, 71, 150 (p = 6) for d = 1, 2, 3 (Remark 5.3).
- **Theorem 1.4** (computer-assisted). Thirteen pairs (p,d): Step 1, for n ≥ n_0 (n_0 = 4, 20, 54, 121 for p = 4,
  d = 1..4; 3, 8, 16, 26 for p = 6; 3, 6, 11 for p = 8; 3, 5 for p = 10), the polynomial
  P(n) = n^p(n−1)(L(n) − E_iid(n)) has P(n_0 + t) with nonnegative coefficients and P(n_0) > 0; Step 2, exact
  values of E_lat − E_iid for 3 ≤ n < n_0. The cases (4,1) and (6,1) are done without a computer in Section 6
  (E_iid(n,1,4) = (3n−1)/(30n³), E_iid(n,1,6) = (9n² − 8n + 2)/(84n⁵)).
- **Conjecture 1.5**, **Remarks 7.1–7.3** (evidence; E disc_p; other readings) and five open questions.

## Computations (programs and outputs in reproducibility/)
The proofs of Sections 2–5 do not depend on computations. Theorem 1.4 is computer-assisted; both of its steps are
exact computations that were carried out by several programs written independently of each other.
- **The exact program of the note** (`writing_stage/check_note.py`, standard library, about 13 s; TOTAL
  failures: 0; 53 checks). φ_α(S) from the definition, by cutting the (z, Δ)-square, against the gap formula on
  40 random (S, α) and against Proposition 4.1 and Remark 4.2(a); Lemma 2.1 on 30 random rational point sets;
  E_lat(n,1,p) (p = 4, n ≤ 8 and n = 10, 20; p ∈ {2,6,8,10,30}, n ≤ 3) and E_lat(n,2,4) (n ≤ 4) by a direct
  integration that uses none of Lemmas 2.1–2.3, against the expansion; Theorems 1.1 and 1.2 (d ≤ 12); the values
  of Table 2; Theorem 5.1 and Proposition 5.2 on exact values; T_p and N(d,p); the threshold certificate of
  Step 1 for all thirteen pairs (and n_0 = 245, 469 for (4,5), (4,6)), with P(n_0 − 1) ≤ 0; the cases (4,1) and
  (6,1) of Section 6; seven negative controls.
- **Programs with which the results were first obtained** (`original/`): E_lat − E_iid strictly positive in 378
  cases for p = 4 (n ≤ 65, d ∈ {1,2,3,4,6,10}), 118 cases for p = 4, d = 4 (n ≤ 120), 92 for p = 6 (n ≤ 25,
  d ≤ 4), 24 for p = 8 (n ≤ 10, d ≤ 3) and 8 for p = 10 (n ≤ 6, d ≤ 2), plus 84, 35 and 20 cases of the generic
  rational code; the thresholds n_0 by exact real-root isolation; a direct sympy integration in d = 1 without
  Lemmas 2.1 and 2.3 (p = 4, n = 2..6; p = 6, n = 3..5); Monte Carlo in d = 2 (n = 3: 0.0087898 and 0.0087777 ±
  0.000011 against 6829/777600 = 0.0087822; n = 5: 0.0039744 ± 0.0000095 against 206257/51840000 = 0.0039787).
- **Verification run A** (`verification_run_A/`): φ from the definition; d = 1 by direct integration (p = 4,
  n ≤ 7; p = 6, n ≤ 5; p = 8, n ≤ 4; p = 10, n = 3; p = 2, n ≤ 5); 52 cases with all index tuples enumerated (every
  p = 2 and every n ≤ 2 case an equality, every case with n ≥ 3 and p ≥ 4 strict); Step 2 of Theorem 1.4 for
  p = 4, d ≤ 4, n ≤ 120 (4 × 118 cases), p = 6, d ≤ 4, n ≤ 25 (4 × 23), p = 8, d ≤ 3, n ≤ 10 (3 × 8), p = 10,
  d ≤ 2, n ≤ 6 (2 × 4), all positive; Step 1 by the coefficient certificate and a Sturm count, with E_iid computed
  by set partitions; Monte Carlo in d = 2 and d = 3; E disc_p in d = 1 by quadrature (n ≤ 5) and Monte Carlo
  (n ≤ 300).
- **Verification run B** (`verification_run_B/`): d = 1 by exact polygon integration without any lemma (n ≤ 14 for
  p = 4, ≤ 10 for p = 6, ≤ 8 for p = 8, ≤ 7 for p = 10), d = 2, 3 by a product decomposition (15 cases); the closed
  forms of Theorem 1.2 for d ∈ {1,2,3,5,8,13,20,30}; Step 2 of Theorem 1.4 with its own expansion code (472, 92,
  32, 12 cases with n ≥ 3 for p = 4, 6, 8, 10, all positive), with E_iid in three ways; Step 1 by root isolation
  and a root count up to a Cauchy bound, with minimality; 1,830 further cases (p = 4, d ≤ 40, n ≤ 40; p = 6, 8,
  10 with d up to 20, 16, 12), all positive; Theorem 5.1 for d = 1 and p ≤ 30; 22 negative controls, all
  detected; Lemma 2.1 on 40 random rational point sets; Monte Carlo in d = 1, 2; quadrature of E disc_p in d = 1;
  a re-run of every program of `original/` from a copy, with every recorded output reproduced.
- **Re-runs at the writing** (2026-10-14, from an extracted copy of the package, Python 3.13.5, sympy 1.14.0,
  numpy 2.5.2): `run_quick.sh` re-ran 35 recorded outputs of all four parts in about 3 minutes, and all are
  identical to the recorded files up to fields that record running times; the three slow exact programs of
  Step 2 for p = 4, n ≤ 120 (about 95–130 s each) were run separately, and their four outputs are identical too
  (`reproducibility/RERUN_LOG.txt`).
- **Verification run of the final text** (`independent_run_2/`, its own programs; see the section of that name
  below): Step 2 of Theorem 1.4 with the full expansion (472, 92, 32, 12 cases with n ≥ 3 for p = 4, 6, 8, 10, all
  positive; values identical, as fractions, to run B's 638 recorded values), Step 1 for all pairs, and a re-run of
  the package from the archive (35 of 35 quick outputs and 4 of 4 slow outputs identical up to running times).
- **Numerical results that are not proofs.** E disc_p (no p-th power): lattices smaller than independent points in
  every computed case (d = 1: n = 3, p = 4: 0.2601443 against 0.2681902 by quadrature; Monte Carlo up to n = 300;
  d = 2, 3 by Monte Carlo). Order of E_lat: n E_lat(n,1,4) = 0.02875, 0.02813, 0.02796, 0.02791, 0.02790 for
  n = 10, 20, 40, 80, 120 (exact values), suggesting E_lat ≍ 1/n; no upper bound is proved.

## Independent verification runs
The results were first obtained, with proofs and with the programs in `original/`. Two independent verification
runs, both AI-assisted, followed on 2026-10-14; each wrote its own programs from the definitions, and neither saw
the report of the other. Run A concentrated on the statement and the proofs, run B on independent computation and
on the literature. Both examined the first written version of the results; the note was written afterwards.

| Item | Run A | Run B |
|---|---|---|
| Statement and definitions against the source (pp. 696–699, read in full) | CONFIRMED; the record's wording ("expected Lp discrepancy") differs from the source | CONFIRMED; the same observation |
| Lemmas 2.1–2.3, Corollary 2.4, Proposition 2.5 | CONFIRMED (re-derived; φ also from the definition) | CONFIRMED (re-derived; Lemma 2.1 also on 40 random point sets) |
| Theorem 1.1 (p = 2; n ≤ 2) | CONFIRMED | CONFIRMED |
| Proposition 4.1, Remark 4.2(a) | CONFIRMED (re-derived; by program) | CONFIRMED (by program for α ≤ 12; Remark 4.2(a) proved) |
| Theorem 1.2 (n = 3, 4; the counterexample 19/1620 > 16/1620) | CONFIRMED (re-derived, also d = 2; by a method without Lemmas 2.1 and 2.3) | CONFIRMED (re-derived; exact closed forms for eight values of d; direct integration in d = 1, 2, 3) |
| Theorem 5.1, Proposition 5.2, Theorem 1.3 | CONFIRMED (each step re-derived; N(d,p) recomputed) | CONFIRMED (each step re-derived; no violation of the two bounds in 2,438 exact cases; p up to 30) |
| Theorem 1.4, Step 2 | CONFIRMED (own code; all 13 pairs) | CONFIRMED (own code; all 13 pairs) |
| Theorem 1.4, Step 1 | CONFIRMED (a different certificate: coefficients of P(n_0 + t) and a Sturm count) | CONFIRMED (root isolation, root count up to a Cauchy bound, minimality) |
| Remark 4.2(b) (closed form of φ_2 for triples) | verified for b ≤ 60, not proved | verified for b ≤ 119, not proved |
| Conjecture 1.5 | TESTED only | TESTED only; 1,830 further exact cases, all positive |
| Reading E disc_p (record's words, literally) | found the opposite inequality in all computed cases (d = 1, 2, 3) | the same (d = 1; quadrature and Monte Carlo) |
| Literature and novelty | three works citing the report found; no answer found | random Weyl sampling and Dolgopyat–Fayad found; Hinrichs 2013 and Novak–Woźniakowski Vol. II not accessible; no answer found |
| Overall | CONFIRMED, with fixes | CONFIRMED, with fixes |

Neither run found a wrong theorem, proposition or lemma, or a gap in a proof.

**Corrections required by the runs**, all applied in the note:
1. (A, B) State the answer for E disc_p^p (p even) exactly as in the source; the record's wording "expected Lp
   discrepancy" is not a faithful restatement; add that for E disc_p the computed cases go the other way:
   abstract, Section 1.1 ("The corpus record"), Remark 7.2, "Scope and priority".
2. (A) Qualify "no constant-factor version" (fixed d and p, n → ∞) and do not claim that the averaging route of the
   source fails for n polynomial in d: Theorem 1.3, Section 1.3, abstract, "Scope and priority".
3. (A, B) Add the works citing the report (Gnewuch 2005, Steinerberger 2010, Hinrichs–Weyhausen 2012, Hinrichs
   2013) and, from run B, random Weyl sampling (Sugita–Takanobu 2000, Sugita 2002) for the case p = 2 and
   Dolgopyat–Fayad 2020: Section 1.4, Remark 3.1, "Scope and priority".
4. (A) Novak–Pillichshammer: their footnote 3 concerns the symmetrized bound; "nothing on this question" rather
   than "does not discuss lattices": Section 1.4.
5. (A) Wnuk–Gnewuch–Hebbinghaus: N prime; their term "not C_0^d-negatively dependent": Section 1.4.
6. (A) The remark on badly approximable generators is restricted to d = 1: Section 1.3.
7. (A, B) The older numbering in the docstrings and in `out_bounds_check.txt`: a correspondence table in
   `reproducibility/README.md` (the recorded files are kept as they were produced).
8. (A, B) Journal versions and titles: Dick–Hinrichs–Pillichshammer (Arch. Math. 115 (2020) 67–78),
   Wnuk–Gnewuch–Hebbinghaus (De Gruyter 2020, 43–68), Du–Dick (journal title), Dick–Pillichshammer and Du–Dick
   (J. Complexity 95), Heinrich–Novak–Wasilkowski–Woźniakowski (doi:10.4064/aa96-3-7); all checked with Crossref.
9. (B) List as not read: Hinrichs 2013 (MCQMC 2012, PROMS 65, 129–172) and Novak–Woźniakowski, Tractability
   Vol. II, Chapter 9 and Appendix D; and Hinrichs–Weyhausen 2012 (the publisher refused automated access):
   "What was read" in Section 8.

**Changes made when the note was written, after the runs.**
- The statements were renumbered (correspondence in `reproducibility/README.md`) and all proofs written out in
  the present form; the proofs are those of the first written version with the corrections above.
- Added: the proofs without a computer of the cases (4,1) and (6,1) of Theorem 1.4 (Section 6; E_iid(n,1,4) = (3n−1)/(30n³),
  E_iid(n,1,6) = (9n² − 8n + 2)/(84n⁵) and two polynomial inequalities); the derivation of φ_1({0,1,2,4}) = 85/432
  in Proposition 4.1(c); Step 1 of Theorem 1.4 in the form of the coefficient certificate (the certificate of
  run A); Remark 7.3 (other readings).
- The exact program of the note, `writing_stage/check_note.py`, was written to check the statements as printed,
  with its own implementation (standard library only) of φ, of the direct integration in d = 1 and d = 2, of the
  expansion and of the threshold certificate.

## Verification run of the final text (2026-10-14)
After the note was written, a further independent verification run, AI-assisted, examined its text. Its programs
(in `reproducibility/independent_run_2/`, with a README) were written from the definitions without reading those of
the other folders.
- **Statement.** The source was fetched again (the same file, sha256 `2f02cd58…6f3d3e`) and read in full, with the
  formulas checked on rendered images of pp. 697–699; the question, the definitions, the role of even p and of the
  p-th power, and the corpus record's different wording are treated correctly in the note.
- **Proofs.** All lemmas, propositions and theorems re-derived line by line, including every gap list and integral
  of Proposition 4.1, the constants of the proof of Theorem 1.2, both steps of Theorem 5.1, Proposition 5.2 and the
  polynomials f and g of Section 6. No error, no gap.
- **Cited results.** Re-read: Novak–Pillichshammer arXiv v1 (definition of the average L_p discrepancy; Section 4
  with footnote 3), Gnewuch 2005 §1, Steinerberger 2010 §1, Wnuk–Gnewuch–Hebbinghaus arXiv v3 (Theorem 4.6 and the
  remark after it), Dick–Hinrichs–Pillichshammer arXiv v1 (Remark 2), the arXiv abstracts of Dolgopyat–Fayad,
  Dick–Pillichshammer and Du–Dick, and the zbMATH reviews 0962.65003 and 0996.65002. All statements agree. Open
  versions of the unread sources: none for Hinrichs 2013 and Novak–Woźniakowski Vol. II; Hinrichs–Weyhausen 2012 is
  in Elsevier's open archive, but the site refused automated access again.
- **Computations** (all exact unless stated): gap formula against a brute-force double integral; Lemma 2.1 against
  a grid-cell integration; E_lat(n,1,p) directly without any lemma (p = 4, n ≤ 8; p = 2, 6, 8, 10, small n);
  Theorems 1.1 and 1.2 (d ≤ 12); Step 2 of Theorem 1.4 with E_iid from the moment–cumulant recursion (p = 4: d ≤ 4,
  n ≤ 120; p = 6: d ≤ 4, n ≤ 25; p = 8: d ≤ 4, n ≤ 10; p = 10: d ≤ 3, n ≤ 6; and p = 4, d = 10, 20, n ≤ 12), all
  positive; Step 1 for the 13 pairs and (4,5), (4,6): certificate, P(n_0 − 1) ≤ 0, least certified n_0 equal to
  the table, largest real root in [n_0 − 1, n_0); T_p, all values of N(d,p), Table 2, Remarks 4.2, 5.3, 7.1 and the
  ratios of Section 1.3; Theorem 5.1 and Proposition 5.2 on all exact values; Remark 7.2 by its own quadrature
  (numerical). Its exact values are identical to all 638 values recorded by run B.
- **Re-run of the package** from an extracted copy of `source.zip`: `run_quick.sh` 35/35 identical; the three slow
  exact programs 4/4 identical (up to running times).
- **Literature** (arXiv API, Crossref for all DOIs, OpenAlex, zbMATH, one web search): nothing answering the
  question; OpenAlex still lists three works citing the report.
- **Corrections required and made:** (1) the count in Remark 7.1 and Section 1.2: 2,354 distinct triples (n,d,p)
  with n ≥ 3, p ≥ 4 instead of "more than 2,400" (which counted overlapping ranges twice); (2) the two Monte Carlo
  runs with 2·10^6 samples quoted in Section 8 belong to the programs of `original/`, not to the verification runs;
  (3) Hinrichs–Weyhausen 2012: open archive, automated access refused; (4) the record of this run (Section 6,
  Table 1, Section 8, this report, `reproducibility/README.md`). Also added: Novak–Pillichshammer, Section 4, as a
  source for the use of the average L_p discrepancy by Heinrich, Novak, Wasilkowski and Woźniakowski.

## Relation to the literature, novelty and scope
- **Searches** (2026-10-14: when the results were first obtained, by runs A and B, in part again at the
  writing, and by the verification run of the final text): arXiv API (Kronecker, lattice, random shift, random generator, average or expected L_p discrepancy,
  pairwise independence, random Weyl sampling, authors Novak and Hinrichs), zbMATH API, Crossref, the lists of
  works citing the report in OpenAlex and Semantic Scholar, and nine general web searches in all. No answer to
  the question, in either direction, was found.
- **Works citing the report that were found:** Gnewuch, Electron. J. Combin. 12 (2005) R54 (read: the
  symmetrization approach for independent points; it suspects an error in the calculations of the report);
  Steinerberger, Electron. J. Combin. 17 (2010) R106 (read: limits of the average L_p discrepancies of independent
  points); Hinrichs–Weyhausen, J. Complexity 28 (2012) 425–439 (in the publisher's open archive; automated access was refused during the verification runs; read on 14 October 2026 after the runs: average L_p discrepancies of independent points only, no statement on lattices or shifted point sets); Novak–Pillichshammer, arXiv:2512.08364 (read in
  version 1: footnote 3 says that the version of the symmetrized bound in the report contains a minor error;
  randomized lattice rules appear only in passing; a journal article of the same authors with the same references
  appeared in J. Complexity 95 (2026) 102028, not read); Hinrichs, MCQMC 2012, Springer PROMS 65 (2013) 129–172
  (closed access, not read; according to its zbMATH review, read by run B, it has sections on average discrepancy
  and on the star discrepancy of structured sets, and it may contain a remark on the question).
- **Related work cited in the note:** random Weyl sampling (Sugita–Takanobu, Monte Carlo Methods Appl. 6 (2000)
  27–48; Sugita, J. Comput. Appl. Math. 139 (2002) 1–8; zbMATH reviews read): pairwise independent samples
  {jz + Δ}, so the case p = 2 is known in substance. Dolgopyat–Fayad, Publ. Math. IHÉS 132 (2020) 293–352
  (abstract read): for z uniform on the torus, the discrepancy of {jz} in a random box, normalized by (ln N)^d, has
  a Cauchy limit law; the same phenomenon (rare near-rational generators) in another normalization, without a
  comparison with independent points. Wnuk–Gnewuch–Hebbinghaus, De Gruyter 2020, 43–68 (arXiv v3 read in the
  cited passages): randomly shifted and jittered rank-1 lattices with a generator from a finite grid, N prime,
  are pairwise negatively dependent (their Theorem 4.6) and, for d ≥ 2 and N ≥ 3, in general not
  C_0^d-negatively dependent. Dick–Pillichshammer and Du–Dick, J. Complexity 95 (2026) 102032 and 102043
  (abstracts read): unions of randomly (digitally) shifted Korobov lattice point sets with finite generators.
  Dick–Hinrichs–Pillichshammer, Arch. Math. 115 (2020) 67–78 (Remark 2 read): Open Problem 40 of
  Novak–Woźniakowski, Vol. II, p. 57.
- **Also searched, nothing on the question:** Novak, "Optimal algorithms for numerical integration: recent results
  and open problems", MCQMC 2022, Springer PROMS (2024) 105–131 (arXiv:2307.06787; full text searched by both
  runs); not cited in the note.
- **Not accessible:** Novak–Woźniakowski, Tractability of Multivariate Problems, Vol. II (EMS 2010,
  doi:10.4171/084), Chapter 9 "Discrepancy and Integration" (pp. 1–70) and Appendix D "List of Open Problems"
  (pp. 627–631; contents page read): a remark on the question there cannot be excluded. Hinrichs 2013 (see above). The content of Sinescu 2009, named in the record's literature note, was not
  read (bibliographic data checked).
- **Assessment.** No source answers the question. The negative answer is elementary once the progression event
  is noticed, and two possibly relevant sources could not be read; the note therefore says that no answer was
  found in the literature accessible to us, that a search that finds nothing is not a proof of novelty, and it
  claims no priority. New, to our knowledge and relative to the literature found: the negative answer, the closed
  forms of Theorem 1.2, Theorem 1.3 with the explicit N(d,p), and Theorem 1.4.
- **Scope.** The answer concerns E disc_p^p with even p, as in the source. Not settled: Conjecture 1.5, the order
  of E_lat, the comparison of E disc_p, the regime n = poly(d), and the upper bound for the star discrepancy of
  lattices asked in the last sentence of the source.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
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