Finite-State Dimension and Real Arithmetic

David Doty; Jack H. Lutz; Satyadev Nandakumar

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arXiv · cs/0602032

Finite-State Dimension and Real Arithmetic

David Doty·Jack H. Lutz·Satyadev Nandakumar

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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 alpha, the base-k expansions of alpha, q+alpha, and q*alpha 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.

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David Doty, Jack H. Lutz, Satyadev Nandakumar. 2006-02-09. Finite-State Dimension and Real Arithmetic. https://arxiv.org/abs/cs/0602032

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