Finite-State Dimension and Real Arithmetic

  • David Doty
  • Jack H. Lutz
  • Satyadev Nandakumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4051)


We use entropy rates and Schur concavity to prove that, for every integer k ≥2, every nonzero rational number q, and every real number α, the base-k expansions of α, q + α, and all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall’s 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.


Shannon Entropy Nonzero Entry Normal Base Normal Sequence Kolmogorov Complexity 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • David Doty
    • 1
  • Jack H. Lutz
    • 1
  • Satyadev Nandakumar
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

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