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Finite-State Dimension and Real Arithmetic



ddoty,lutz,satyadev} at cs dot iastate dot edu 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 ff, the base-k expansions of ff, q + ff, and qff 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. 1 Introduction The finite-state dimension of a sequence S over a finite alphabet \Sigma is an as-ymptotic measure of the density of information in S as perceived by finite-stateautomata. This quantity, denoted di

Year: 2008
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