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Finite-State Dimension and the Davenport-Erdős Theorem

Authors Joe Clanin , Matthew Rayman

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LIPIcs.MFCS.2026.38.pdf
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ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • Normal numbers
  • finite-state dimension
  • polynomials

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.MFCS.2026.38

Author Details

Joe Clanin
  • Department of Computer Science, Iowa State University, Ames, IA, USA
Matthew Rayman
  • Department of Computer Science, Iowa State University, Ames, IA, USA

Funding

This research was supported in part by National Science Foundation research grants 1900716 and 1545028.

Acknowledgements

The authors thank Jack Lutz for many helpful discussions as well as the anonymous reviewers and Wolfgang Merkle for their detailed comments.

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