Document Open Access Logo

A Weyl Criterion for Finite-State Dimension and Applications

Authors Jack H. Lutz , Satyadev Nandakumar , Subin Pulari

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2023.65.pdf
  • Filesize: 0.77 MB
  • 16 pages

Document Identifiers

Author Details

Jack H. Lutz
  • Department of Computer Science, Iowa State University, Ames, IA, USA
Satyadev Nandakumar
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, U.P., India
Subin Pulari
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, U.P., India

Funding

  • Lutz, Jack H.: This author’s research was supported in part by National Science Foundation grant 1900716.

Acknowledgements

The authors would like to thank Michael Hochman for technical clarifications regarding his paper [Hochman, 2014]. We also thank the anonymous reviewers for their valuable comments and suggestions.

Cite AsGet BibTex

Jack H. Lutz, Satyadev Nandakumar, and Subin Pulari. A Weyl Criterion for Finite-State Dimension and Applications. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 65:1-65:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.MFCS.2023.65

Abstract

Finite-state dimension, introduced early in this century as a finite-state version of classical Hausdorff dimension, is a quantitative measure of the lower asymptotic density of information in an infinite sequence over a finite alphabet, as perceived by finite automata. Finite-state dimension is a robust concept that now has equivalent formulations in terms of finite-state gambling, lossless finite-state data compression, finite-state prediction, entropy rates, and automatic Kolmogorov complexity. The 1972 Schnorr-Stimm dichotomy theorem gave the first automata-theoretic characterization of normal sequences, which had been studied in analytic number theory since Borel defined them in 1909. This theorem implies, in present-day terminology, that a sequence (or a real number having this sequence as its base-b expansion) is normal if and only if it has finite-state dimension 1. One of the most powerful classical tools for investigating normal numbers is the 1916 Weyl’s criterion, which characterizes normality in terms of exponential sums. Such sums are well studied objects with many connections to other aspects of analytic number theory, and this has made use of Weyl’s criterion especially fruitful. This raises the question whether Weyl’s criterion can be generalized from finite-state dimension 1 to arbitrary finite-state dimensions, thereby making it a quantitative tool for studying data compression, prediction, etc. i.e., Can we characterize all compression ratios using exponential sums?. This paper does exactly this. We extend Weyl’s criterion from a characterization of sequences with finite-state dimension 1 to a criterion that characterizes every finite-state dimension. This turns out not to be a routine generalization of the original Weyl criterion. Even though exponential sums may diverge for non-normal numbers, finite-state dimension can be characterized in terms of the dimensions of the subsequence limits of the exponential sums. In case the exponential sums are convergent, they converge to the Fourier coefficients of a probability measure whose dimension is precisely the finite-state dimension of the sequence. This new and surprising connection helps us bring Fourier analytic techniques to bear in proofs in finite-state dimension, yielding a new perspective. We demonstrate the utility of our criterion by substantially improving known results about preservation of finite-state dimension under arithmetic. We strictly generalize the results by Aistleitner and Doty, Lutz and Nandakumar for finite-state dimensions under arithmetic operations. We use the method of exponential sums and our Weyl criterion to obtain the following new result: If y is a number having finite-state strong dimension 0, then dim_FS(x+qy) = dim_FS(x) and Dim_FS(x+qy) = Dim_FS(x) for any x ∈ ℝ and q ∈ ℚ. This generalization uses recent estimates obtained in the work of Hochman [Hochman, 2014] regarding the entropy of convolutions of probability measures.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Information theory
Keywords
  • Finite-state dimension
  • Finite-state compression
  • Weyl’s criterion
  • Exponential sums
  • Normal numbers

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Christoph Aistleitner. On modifying normal numbers. Unif. Distrib. Theory, 6(2):49-58, 2011. Google Scholar
  2. Krishna B Athreya, John M Hitchcock, Jack H Lutz, and Elvira Mayordomo. Effective strong dimension in algorithmic information and computational complexity. SIAM journal on computing, 37(3):671-705, 2007. Google Scholar
  3. Verónica Becher and Pablo Ariel Heiber. Normal numbers and finite automata. Theoretical Computer Science, 477:109-116, 2013. Google Scholar
  4. Patrick Billingsley. Convergence of probability measures. John Wiley & Sons, 2013. Google Scholar
  5. Chris Bourke, John M Hitchcock, and NV Vinodchandran. Entropy rates and finite-state dimension. Theoretical Computer Science, 349(3):392-406, 2005. Google Scholar
  6. Y. Bugeaud. Distribution Modulo 1 and Diophantine Approximation. Pure and Applied Mathematics. Cambridge University Press, 2012. Google Scholar
  7. J. W. S. Cassels. On a problem of Steinhaus about normal numbers. Colloq. Math., 7:95-101, 1959. URL: https://doi.org/10.4064/cm-7-1-95-101.
  8. D. G. Champernowne. Construction of decimals normal in the scale of ten. J. London Math. Soc., 2(8):254-260, 1933. Google Scholar
  9. T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, Inc., New York, N.Y., 1991. Google Scholar
  10. Jack J Dai, James I Lathrop, Jack H Lutz, and Elvira Mayordomo. Finite-state dimension. Theoretical Computer Science, 310(1-3):1-33, 2004. Google Scholar
  11. H. Davenport and P. Erdös. Note on normal decimals. Canad. J. Math., 4:58-63, 1952. URL: https://doi.org/10.4153/cjm-1952-005-3.
  12. D. Doty, J. H. Lutz, and S. Nandakumar. Finite state dimension and real arithmetic. In Proceedings of the 33rd International Colloquium on Automata, Logic and Programming, 2006. Google Scholar
  13. Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York, 2010. URL: https://doi.org/10.1007/978-0-387-68441-3.
  14. Manfred Einsiedler and Thomas Ward. Ergodic theory with a view towards number theory, volume 259 of Graduate Texts in Mathematics. Springer-Verlag London, Ltd., London, 2011. URL: https://doi.org/10.1007/978-0-85729-021-2.
  15. M. Feder. Gambling using a finite state machine. IEEE Transactions on Information Theory, 37:1459-1461, 1991. Google Scholar
  16. Gerald B Folland. A course in abstract harmonic analysis, volume 29. CRC press, 2016. Google Scholar
  17. Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy. Ann. of Math. (2), 180(2):773-822, 2014. URL: https://doi.org/10.4007/annals.2014.180.2.7.
  18. A. I. Khinchin. Mathematical foundations of information theory. Dover Publications, Inc., New York, N. Y., 1957. Translated by R. A. Silverman and M. D. Friedman. Google Scholar
  19. Alexander Kozachinskiy and Alexander Shen. Two characterizations of finite-state dimension. In International Symposium on Fundamentals of Computation Theory, pages 80-94. Springer, 2019. Google Scholar
  20. L. Kuipers and H. Niederreiter. Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Google Scholar
  21. Jack H. Lutz. Dimension in complexity classes. SIAM J. Comput., 32(5):1236-1259, 2003. URL: https://doi.org/10.1137/S0097539701417723.
  22. Manfred G Madritsch and Bill Mance. Construction of μ-normal sequences. Monatshefte für Mathematik, 179(2):259-280, 2016. Google Scholar
  23. Joseph S. Miller. Extracting information is hard: a Turing degree of non-integral effective Hausdorff dimension. Adv. Math., 226(1):373-384, 2011. URL: https://doi.org/10.1016/j.aim.2010.06.024.
  24. I. Niven. Irrational Numbers. Carus Mathematical Monographs, 1956. Google Scholar
  25. A. Rényi. On the dimension and entropy of probability distributions. Acta Math. Acad. Sci. Hungar., 10:193-215 (unbound insert), 1959. URL: https://doi.org/10.1007/BF02063299.
  26. Walter Rudin. Fourier analysis on groups. Interscience Tracts in Pure and Applied Mathematics, No. 12. Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. Google Scholar
  27. Wolfgang M. Schmidt. Über die Normalität von Zahlen zu verschiedenen Basen. Acta Arith., 7:299-309, 1961/62. URL: https://doi.org/10.4064/aa-7-3-299-309.
  28. C. P. Schnorr and H. Stimm. Endliche Automaten und Zufallsfolgen. Acta Informatica, 1:345-359, 1972. Google Scholar
  29. A. N. Shiryaev. Probability. Graduate Texts in Mathematics v.95. Springer, 2 edition, 1995. Google Scholar
  30. Elias M. Stein and Rami Shakarchi. Fourier analysis, volume 1 of Princeton Lectures in Analysis. Princeton University Press, Princeton, NJ, 2003. An introduction. Google Scholar
  31. D. D. Wall. Normal Sequences. PhD thesis, University of California, Berkeley, 1949. Google Scholar
  32. Hermann Weyl. Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann., 77(3):313-352, 1916. URL: https://doi.org/10.1007/BF01475864.
  33. Lai-Sang Young. Dimension, entropy and Lyapunov exponents. Ergodic theory and dynamical systems, 2(1):109-124, 1982. Google Scholar
  34. J. Ziv and A. Lempel. Compression of individual sequences via variable rate coding. IEEE Transaction on Information Theory, 24:530-536, 1978. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact