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DOI: 10.4230/LIPIcs.MFCS.2026.38

Finite-State Dimension and the Davenport-Erdős Theorem

Authors: Joe Clanin and Matthew Rayman

Published in: LIPIcs, Volume 386, 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)

Abstract

A 1952 result of Davenport and Erdős states that if p is an integer-valued polynomial, then the real number 0.p(1)p(2)p(3)… is Borel normal in base ten. A later result of Nakai and Shiokawa extends this result to polynomials with arbitrary real coefficients and all bases b ≥ 2. It is well-known that finite-state dimension, a finite-state effectivization of the classical Hausdorff dimension, characterizes the Borel normal sequences as precisely those sequences of finite-state dimension 1. For an infinite set A of natural numbers, and a base b ≥ 2, the base-b Copeland-Erdős sequence of A, CE_b(A), is the infinite sequence obtained by concatenating the base-b expansions of the numbers in A in increasing order. In this work we investigate the possible relationships between the finite-state dimensions of CE_b(A) and CE_b(p(A)) where p is a polynomial. We show that, if the polynomial is permitted to have arbitrary real coefficients, then for any s,s^′ in the unit interval, there is a set A of natural numbers and a linear polynomial p so that the finite-state dimensions of CE_b(A) and CE_b(p(A)) are s and s^′ respectively. The corresponding result for strong finite-state dimension is also shown. We demonstrate that linear polynomials with rational coefficients do not change the finite-state dimension of any Copeland-Erdős sequence, but there exist polynomials with rational coefficients of every larger integer degree that change the finite-state dimension of some sequence. We also prove the surprising fact that there exist sets A and integer-valued monomials p such that CE_b(A) is normal, but CE_b(p(A)) has finite-state dimension strictly less than one.

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Joe Clanin and Matthew Rayman. Finite-State Dimension and the Davenport-Erdős Theorem. In 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 386, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{clanin_et_al:LIPIcs.MFCS.2026.38,
  author =	{Clanin, Joe and Rayman, Matthew},
  title =	{{Finite-State Dimension and the Davenport-Erd\H{o}s Theorem}},
  booktitle =	{51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)},
  pages =	{38:1--38:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-442-0},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{386},
  editor =	{Kouck\'{y}, Michal and Petrișan, Daniela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2026.38},
  URN =		{urn:nbn:de:0030-drops-274195},
  doi =		{10.4230/LIPIcs.MFCS.2026.38},
  annote =	{Keywords: Normal numbers, finite-state dimension, polynomials}
}

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DOI: 10.4230/LIPIcs.STACS.2026.75

Effective Versions of Strong Measure Zero

Authors: Matthew Rayman

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

Effective versions of strong measure zero sets are developed for various levels of complexity and computability. It is shown that the sets can be equivalently defined using a generalization of supermartingales called odds supermartingales, success rates on supermartingales, predictors, and coverings. We show Borel’s conjecture that a set has strong measure zero if and only if it is countable holds in the time and space bounded setting. At the level of computability this does not hold. We show the computable level contains sequences at arbitrary levels of the hyperarithmetical hierarchy. This is done by proving a correspondence principle yielding a condition for the sets of computable strong measure zero to agree with the classical sets of strong measure zero. An algorithmic version of strong measure zero using lower semicomputability is defined. We show that this notion is equivalent to the set of NCR reals studied by Reimann and Slaman, thereby giving new characterizations of this set. Effective strong packing dimension zero is investigated by requiring success with respect to the limit inferior instead of the limit superior. It is proven that every sequence in the corresponding algorithmic class is decidable. At the level of computability, the sets coincide with a notion of weak countability that we define.

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Matthew Rayman. Effective Versions of Strong Measure Zero. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{rayman:LIPIcs.STACS.2026.75,
  author =	{Rayman, Matthew},
  title =	{{Effective Versions of Strong Measure Zero}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{75:1--75:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.75},
  URN =		{urn:nbn:de:0030-drops-255648},
  doi =		{10.4230/LIPIcs.STACS.2026.75},
  annote =	{Keywords: Strong measure zero, NCR, Effective fractal dimensions, Borel’s Conjecture, Hausdorff dimension, Packing dimension}
}

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Finite-State Dimension and the Davenport-Erdős Theorem

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Effective Versions of Strong Measure Zero

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