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Normal numbers from independent processes

Published online by Cambridge University Press:  19 September 2008

J. Feldman  and
M. Smorodinsky
J. Feldman
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA
M. Smorodinsky
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel
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Extract

In 1960 Schmidt [S] showed that if p and q are not powers of the same integer, i.e., if log q/log p is irrational, then for certain special measures μ on [0,1), invariant under S:x ↦ px (mod 1), μ-almost every x is normal to the base q. The measures considered in [S] were similar to Cantor-Lebesgue measure: namely, under μ the p-digit process was a special i.i.d. process where for some k ≥ 2 the elements of a certain k-element subset of the p-digits assumed probability 1/k each. The proof was fairly complicated, and did not seem to yield much more (see Keane and Pearce [K] for another proof).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 4 , December 1992 , pp. 707 - 712
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

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