{
  "schema_version": 1,
  "problem_number": "OWR-15986-005",
  "title": "On a Question of Becher and Slaman: For Kurtz Random x the Discrepancy of (2^n x) Is Not O((log N)/N)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In Oberwolfach Report 1/2018, Becher and Slaman ask whether some Martin-Löf random real x has discrepancy D_N((2^n x)_{n≥1}) = O((log N)/N) for all large N. Read as asking for constants C, N_0 with N D_N ≤ C log N for all N ≥ N_0, the question has a negative answer, also for Schnorr, computably and Kurtz random x. More precisely, if x is Kurtz random and g is computable with limsup_N g(N)/√(N log log N) < 1/√2, for instance any computable g(N) = o(√(N log log N)), then N D_N > g(N) for infinitely many N; in particular limsup_N N D_N/√(N log log N) ≥ 1/√2. We give two short proofs of the negative answer. The first combines Koksma's inequality for e^{2πit}, the law of the iterated logarithm of Erdős and Gál for lacunary exponential sums, and the fact that a null effectively closed set contains no Kurtz random real; it gives the constant 1/(2π) in place of 1/√2. The second replaces these analytic tools by the observation that {2^n x} < 1/2 exactly when the (n+1)st binary digit of x is 0, so that N D_N is at least the deviation from N/2 of a count of zeros; a binomial estimate then gives the negative answer, and Khinchin's law of the iterated logarithm gives the constant 1/√2. For Martin-Löf random x the negative answer follows at once from this digit count and Vovk's law of the iterated logarithm for Martin-Löf random sequences; the note records a short observation from known results, which may be folklore. The question is due to Becher and Slaman, and the tools are classical. We do not treat the variant with \"for infinitely many N\", nor Korobov's question on the existence of a normal number with discrepancy O((log N)/N), which the source connects with the question, nor the other questions of the problem session. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.LO",
    "math.NT",
    "math.PR"
  ],
  "keywords": [
    "discrepancy",
    "lacunary sequences",
    "(2^n x) modulo one",
    "binary expansion",
    "Martin-Löf randomness",
    "Kurtz randomness",
    "effectively closed sets",
    "law of the iterated logarithm",
    "Koksma inequality",
    "Erdős–Gál theorem",
    "Vovk's theorem",
    "normal numbers",
    "Korobov's question",
    "negative answer",
    "Oberwolfach Reports open problems",
    "UnsolvedMath",
    "OWR-15986-005",
    "OWR-15986-006",
    "math.LO",
    "math.NT",
    "math.PR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-15986-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-15986-005/paper.pdf?v=891d1bb8b243",
  "doi": "10.5281/zenodo.23400059",
  "zenodo_record_url": "https://zenodo.org/records/23400059",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "On Question 1 of Becher and Slaman (Oberwolfach Report 1/2018; also the first question of record OWR-15986-006): no Kurtz random real, hence no Martin-Lof random real, has discrepancy O((log N)/N) for the sequence (2^n x); two short proofs from known results, which may be folklore. The other questions of both records remain open.",
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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."
}