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Schnorr randomness and the Lebesgue differentiation theorem


Authors: Noopur Pathak, Cristóbal Rojas and Stephen G. Simpson
Journal: Proc. Amer. Math. Soc. 142 (2014), 335-349
MSC (2010): Primary 03D32, 26A24
DOI: https://doi.org/10.1090/S0002-9939-2013-11710-7
Published electronically: August 27, 2013
MathSciNet review: 3119207
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Abstract: We exhibit a close correspondence between $ L_1$-computable functions and Schnorr tests. Using this correspondence, we prove that a point $ x\in [0,1]^d$ is Schnorr random if and only if the Lebesgue Differentiation Theorem holds at $ x$ for all $ L_1$-computable functions $ f\in L_1([0,1]^d)$.


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Additional Information

Noopur Pathak
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802
Email: noopur.j.pathak@gmail.com

Cristóbal Rojas
Affiliation: Departamento de Matemáticas, Universidad Andres Bello, Santiago, Chile
Email: cristobal.rojas@unab.cl

Stephen G. Simpson
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, State College, Pennsylvania 16802
Email: simpson@math.psu.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11710-7
Received by editor(s): September 29, 2011
Received by editor(s) in revised form: February 17, 2012
Published electronically: August 27, 2013
Additional Notes: The research of the authors was partially supported by NSF grant DMS-0652637 as part of a U.S. National Science Foundation Focused Research Group project on algorithmic randomness.
The authors thank John Clemens for detailed comments on a draft of this paper.
Communicated by: Julia Knight
Article copyright: © Copyright 2013 American Mathematical Society