A strong hot spot theorem
David H. Bailey and Michal\ Misiurewicz
Proc. Amer. Math. Soc. 134 (2006), 2495-2501
Primary 11K16; Secondary 37A30
March 22, 2006
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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.
H. Bailey and Richard
E. Crandall, Random generators and normal numbers, Experiment.
Math. 11 (2002), no. 4, 527–546 (2003). MR 1969644
David H. Bailey, ``A Hot Spot Proof of Normality for the Alpha Constants,'' available at http://crd.lbl.gov/~dhbailey/dhbpapers/alpha-normal.pdf
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- 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)
- David H. Bailey, ``A Hot Spot Proof of Normality for the Alpha Constants,'' available at http://crd.lbl.gov/~dhbailey/dhbpapers/alpha-normal.pdf
- Patrick Billingsley, Ergodic Theory and Information, John Wiley, New York, 1965. MR 0192027 (33:254)
- 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)
- L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley-Interscience, New York, 1974. MR 0419394 (54:7415)
- 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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David H. Bailey
Lawrence Berkeley Laboratory, 1 Cyclotron Road, Berkeley, California 94720
Department of Mathematical Sciences, Indiana University–Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Received by editor(s):
February 1, 2005
Received by editor(s) in revised form:
March 24, 2005
March 22, 2006
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.
Jonathan M. Borwein