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Measures of sets decomposing the simply normal numbers in the unit interval

Part of:Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  26 February 2010

John Slivka and
Norman C. Severo
John Slivka
Affiliation:
Department of Mathematics, State University College, Buffalo, New York 14222, U.S.A.
Norman C. Severo
Affiliation:
Department of Statistics, State University of New York at Buffalo, Buffalo, New York 14214, U.S.A.
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Abstract

For any fixed positive real number ε, any integer b≥2 and any dε{0, 1,…, b−1}, the set of Borel's simply normal numbers to base b in [0, 1] is partitioned into a countable number of sets in eight different ways according to the largest place and the number of places at which the proportion d's to that place in the b-adic expansion of such a number exceeds or is not less than b−1 – ε, or is less than or does not exceed b−1 – ε. For selected values ε, the Lebesgue measures of the sets in these decompositions are given explicitly.

MSC classification

Secondary:11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Information

Type
Research Article
Information
Mathematika , Volume 41 , Issue 1 , June 1994, pp. 164 - 172
Copyright
Copyright © University College London 1994

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References

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