Finite-sate dimension and real arithmetic

Authors:
David Doty;Jack H. Lutz;Satyadev Nandakumar
Affiliations:
Department of Computer Science, Iowa State University, Ames, IA;Department of Computer Science, Iowa State University, Ames, IA;Department of Computer Science, Iowa State University, Ames, IA
Venue:
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part I
Year:
2006

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Abstract

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 qα 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.