A strong hot spot theorem
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- by David H. Bailey and Michał Misiurewicz PDF
- Proc. Amer. Math. Soc. 134 (2006), 2495-2501 Request permission
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.References
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Additional Information
- David H. Bailey
- Affiliation: Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, California 94720
- MR Author ID: 29355
- Email: dhbailey@lbl.gov
- Michał Misiurewicz
- Affiliation: Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
- MR Author ID: 125475
- Email: mmisiure@math.iupui.edu
- 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
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2495-2501
- MSC (2000): Primary 11K16; Secondary 37A30
- DOI: https://doi.org/10.1090/S0002-9939-06-08551-0
- MathSciNet review: 2213726