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Transactions of the American Mathematical Society

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Translating the Cantor set by a random real

Authors: Randall Dougherty, Jack H. Lutz, R. Daniel Mauldin and Jason Teutsch
Journal: Trans. Amer. Math. Soc. 366 (2014), 3027-3041
MSC (2010): Primary 68Q30; Secondary 11K55, 28A78
Published electronically: January 8, 2014
MathSciNet review: 3180738
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Abstract: We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set ``cancels randomness'' in the sense that some of its members, when added to Martin-Löf random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a random Cantor set translate with a given constructive dimension.

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Additional Information

Randall Dougherty
Affiliation: Department of Mathematics, Center for Communications Research–La Jolla, 4320 Westerra Court, San Diego, California 92121

Jack H. Lutz
Affiliation: Department of Computer Science, Iowa State University, Ames, Iowa 50011

R. Daniel Mauldin
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203

Jason Teutsch
Affiliation: Department of Mathematics, Ruprecht-Karls-Universität Heidelberg, D-69120 Heidelberg, Germany

Keywords: Algorithmic randomness, fractal geometry, additive number theory
Received by editor(s): March 25, 2011
Received by editor(s) in revised form: July 11, 2012
Published electronically: January 8, 2014
Additional Notes: The second author’s research was supported by NSF Grants 0652569 and 0728806.
The third author’s research was supported by NSF Grant DMS-0700831.
The fourth author’s research was supported by Deutsche Forschungsgemeinschaft grant ME 1806/3-1.
Article copyright: © Copyright 2014 by the authors

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