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Steinhaus Piercing with Slack: The Exact Asymptotic Constant

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CO — Combinatorics

Contributed by Anton Joha ↗

Version 1 / Oct 14, 2026 / CC BY 4.0

Abstract

Let s(d) be the largest order N for which the first n + d points of a sequence in [0,1) meet every uniform n-cell for all n ≤ N. We prove s(d) = d/(1 − log 2) + O(√d), determining the limit asked for by Anholcer et al. and disproving their conjectured coefficient. For strong piercing, which requires meeting every interval of length 1/n, we obtain an exact finite harmonic capacity and rational constructions. A Dirichlet covering lemma and a packing estimate for disjoint gaps transfer the capacity bound to ordinary grids. We also solve the finite largest-gap minimax problem, establish rigidity for infinite heap refinements, and extend the asymptotic estimate to affine arrival budgets. Ordinary near-extremizers satisfy universal laws for root sizes and first-split times, with a corresponding entropy correction. An explicit finite obstruction shows why root constraints alone do not characterize ordinary piercing.

Provenance statement

During the research and preparation of this work, the author used ChatGPT (OpenAI) and Claude (Anthropic) to assist with organization, clarity, presentation, references and formatting. ChatGPT also assisted with mathematical exploration and the development of enumeration, SAT-encoding, certificate-checking and plotting code for the broader project. Figures were generated from the formulas and data stated in the work. The author reviewed and edited the content and takes responsibility for the manuscript and its mathematical and computational claims.

Tools used

Anthropic
Claude FableVersion Fable 5.1
OpenAI
ChatGPTVersion GPT-6 Astra

References

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  2. E.R. Berlekamp, R.L. Graham, Irregularities in the distributions of finite sequences, J. Number Theory 2 (1970) 152--161. https://doi.org/10.1016/0022-314X(70)90015-6DOI
  3. N.G. de Bruijn, P. Erdős, Sequences of points on a circle, Proc. Sect. Sci. K. Ned. Akad. Wet. Amsterdam 52 (1949) 14--17 = Indag. Math. 11 (1949) 46--49.
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  5. S.V. Konyagin, On irregularity of finite sequences, Proc. Steklov Inst. Math. 314 (2021) 90--95. https://doi.org/10.1134/S0081543821040052DOI
  6. K. Levy, Lower and upper bounds on irregularities of distribution, Integers 20 (2020) A26.
  7. H. Steinhaus, Sto zadań, PWN, Warszawa, 1958; English translation: One Hundred Problems in Elementary Mathematics, Basic Books, New York, 1964; Dover reprint, 1979, Problems 6--7.
  8. M. Warmus, A supplementary note on the irregularities of distributions, J. Number Theory 8 (3) (1976) 260--263. https://doi.org/10.1016/0022-314X(76)90002-0DOI

Version history

  1. v1Submitted by Anton JohaInitial depositCurrentOct 14, 2026