Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. David H. Bailey and Richard E. Crandall, Random generators and normal numbers, Experiment. Math. 11 (2002), no. 4, 527–546 (2003). MR 1969644
  • 2. David H. Bailey, ``A Hot Spot Proof of Normality for the Alpha Constants,'' available at
  • 3. Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
  • 4. Jonathan Borwein and David Bailey, Mathematics by experiment, A K Peters, Ltd., Natick, MA, 2004. Plausible reasoning in the 21st century. MR 2033012
  • 5. L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0419394
  • 6. R. G. Stoneham, On absolute (𝑗,𝜖)-normality in the rational fractions with applications to normal numbers, Acta Arith. 22 (1972/73), 277–286. MR 0318072

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11K16, 37A30

Retrieve articles in all journals with MSC (2000): 11K16, 37A30

Additional Information

David H. Bailey
Affiliation: Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, California 94720

Michal\ Misiurewicz
Affiliation: Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216

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