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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A strong hot spot theorem


Authors: David H. Bailey and Michal\ Misiurewicz
Journal: Proc. Amer. Math. Soc. 134 (2006), 2495-2501
MSC (2000): Primary 11K16; Secondary 37A30
Published electronically: March 22, 2006
MathSciNet review: 2213726
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Abstract: A real number $ \alpha$ is said to be $ b$-normal if every $ m$-long string of digits appears in the base-$ b$ expansion of $ \alpha$ with limiting frequency $ b^{-m}$. We prove that $ \alpha$ is $ b$-normal if and only if it possesses no base-$ b$ ``hot spot''. In other words, $ \alpha$ is $ b$-normal if and only if there is no real number $ y$ such that smaller and smaller neighborhoods of $ y$ are visited by the successive shifts of the base-$ b$ expansion of $ \alpha$ with larger and larger frequencies, relative to the lengths of these neighborhoods.


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Additional Information

David H. Bailey
Affiliation: Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, California 94720
Email: dhbailey@lbl.gov

Michal\ Misiurewicz
Affiliation: Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email: mmisiure@math.iupui.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08551-0
PII: S 0002-9939(06)08551-0
Keywords: Normal numbers
Received by editor(s): February 1, 2005
Received by editor(s) in revised form: March 24, 2005
Published electronically: March 22, 2006
Additional Notes: The submitted manuscript has been authored by a contractor of the U.S. Government under Contract No. DE-AC02-05CH11231. Accordingly, the U.S. Government retains a nonexclusive royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.
Communicated by: Jonathan M. Borwein