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Feasible Analysis, Randomness, and Base Invariance

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Abstract

We show that polynomial time randomness of a real number does not depend on the choice of a base for representing it. Our main tool is an ‘almost Lipschitz’ condition that we show for the cumulative distribution function associated to martingales with the savings property. Based on a result of Schnorr, we prove that for any base r, n⋅log2 n-randomness in base r implies normality in base r, and that n 4-randomness in base r implies absolute normality. Our methods yield a construction of an absolutely normal real number which is computable in polynomial time.

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Acknowledgements

We thank Verónica Becher, Pablo Heiber, Jack Lutz, Elvira Mayordomo and Theodore A. Slaman. We also thank the referee for careful reading and mindful suggestions. This research was partially carried out while the authors participated in the Buenos Aires Semester in Computability, Complexity and Randomness, 2013. Nies was supported by the Marsden fund of New Zealand. Figueira was supported by UBA (UBACyT 20020110100025) and ANPCyT (PICT-2011-0365).

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  1. Departamento de Computación, University of Buenos Aires and CONICET, Pabellón I, Ciudad Universitaria (C1428EGA), Buenos Aires, Argentina

    Santiago Figueira

  2. The University of Auckland, Private Bag 92019, Auckland, New Zealand

    André Nies

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  1. Santiago Figueira
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Correspondence to Santiago Figueira.

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Figueira, S., Nies, A. Feasible Analysis, Randomness, and Base Invariance. Theory Comput Syst 56, 439–464 (2015). https://doi.org/10.1007/s00224-013-9507-7

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