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

DOI:
https://doi.org/10.1090/S0002-9939-06-08551-0

Published electronically:
March 22, 2006

MathSciNet review:
2213726

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Abstract | References | Similar Articles | Additional Information

Abstract: A real number is said to be -normal if every -long string of digits appears in the base- expansion of with limiting frequency . We prove that is -normal if and only if it possesses no base- ``hot spot''. In other words, is -normal if and only if there is no real number such that smaller and smaller neighborhoods of are visited by the successive shifts of the base- expansion of with larger and larger frequencies, relative to the lengths of these neighborhoods.

**1.**David H. Bailey and Richard E. Crandall, ``Random Generators and Normal Numbers,''*Experimental Mathematics***11**(2002), no. 4, 527-546; available at`http://expmath.org/expmath/volumes/11/11.4/pp527_546.pdf`MR**1969644 (2004c:11135)****2.**David H. Bailey, ``A Hot Spot Proof of Normality for the Alpha Constants,'' available at`http://crd.lbl.gov/~dhbailey/dhbpapers/alpha-normal.pdf`**3.**Patrick Billingsley,*Ergodic Theory and Information*, John Wiley, New York, 1965. MR**0192027 (33:254)****4.**Jonathan M. Borwein and David H. Bailey,*Mathematics by Experiment: Plausible Reasoning in the 21st Century*, A K Peters, Natick, MA, 2004. MR**2033012 (2005b:00012)****5.**L. Kuipers and H. Niederreiter,*Uniform Distribution of Sequences*, Wiley-Interscience, New York, 1974. MR**0419394 (54:7415)****6.**R. Stoneham, ``On Absolute -Normality in the Rational Fractions with Applications to Normal Numbers,''*Acta Arithmetica***22**(1973), 277-286. MR**0318072 (47:6621)**

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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:
https://doi.org/10.1090/S0002-9939-06-08551-0

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