{
  "schema_version": 1,
  "problem_number": "OWR-723-005",
  "title": "A Negative Answer to a Question of Novak on Randomly Shifted Lattices and the Expected p-th Power of the L_p Discrepancy",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an Oberwolfach report of 2004, E. Novak (joint work with A. Hinrichs) considered the generalized lattices with shift M_n^{z,Δ} = {jz + Δ mod 1 : 0 ≤ j < n}, with z and Δ independent and uniformly distributed in [0,1]^d, and asked whether, for even p, the expectation of disc_p^p(M_n^{z,Δ}), the p-th power of the L_p discrepancy, is at most the corresponding expectation for n independent uniformly distributed points. We show that the answer is negative. Equality holds for p = 2, because the lattice points are pairwise independent (in substance a known fact, which underlies random Weyl sampling), and for n ≤ 2. For n = 3 and p = 4 the difference of the two expectations is (8/9)·2^(−d)·((2/5)^d − (19/48)^d) > 0 in every dimension d; for d = 1 the two values are 19/1620 and 16/1620. A closed formula shows that n = 4, p = 4 fails in every dimension as well. For every d and every even p ≥ 4 the inequality fails for all n ≥ N(d,p), with N(d,p) explicit: with probability of order 1/n the points form a short arithmetic progression, which gives E disc_p^p ≥ 4/((p+1)^d (p+2)(p+3)(n−1)) for the lattices, against order n^(−p/2) for independent points. For thirteen pairs (p,d), among them p = 4 with d ≤ 4, the inequality fails for every n ≥ 3; this part is computer-assisted (exact rational arithmetic and a certified polynomial inequality). We conjecture failure for all n ≥ 3, d ≥ 1 and even p ≥ 4. The answer concerns the p-th power, as in the source; for the expectation of disc_p itself, which the wording of the corpus record also allows, all computed cases go the other way. The results are asymptotic in n for fixed d and p and do not exclude bounds of the i.i.d. type when n grows polynomially with d. The question is Novak's; no answer to it was found in the literature accessible to us. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.NA",
    "math.NT",
    "math.PR"
  ],
  "keywords": [
    "L_p discrepancy",
    "star discrepancy",
    "average discrepancy",
    "randomly shifted lattices",
    "Kronecker point sets",
    "rank-1 lattices",
    "random Weyl sampling",
    "pairwise independence",
    "quasi-Monte Carlo",
    "negative answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-723-005",
    "math.NA",
    "math.NT",
    "math.PR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-723-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-723-005/paper.pdf?v=edb268431856",
  "doi": "10.5281/zenodo.23372668",
  "zenodo_record_url": "https://zenodo.org/records/23372668",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers E. Novak's question (Oberwolfach Report 13/2004, joint work with A. Hinrichs) in the form of the source, for the expected p-th power of the L_p discrepancy of randomly shifted lattices against independent points: equality for p = 2 and n ≤ 2; the inequality fails for every even p ≥ 4 in every dimension once n ≥ N(d,p), and for all n ≥ 3 in thirteen pairs (p,d) (computer-assisted). For the expectation of the L_p discrepancy itself, which the corpus wording also allows, the computed cases go the other way and nothing is proved. Two closed sources could not be read. Novelty is not certified.",
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      "sha256": "1f96bd15a8f35e8aeac8e3e8b0d71162dd1bbd8ea83a6b53b0645d2fb9e39c11"
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    "verification_report.md": {
      "sha256": "85f01918e5b7b84a77cefe6e11c6eae85d8ac5eef2471174000b51cd346c710e"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}