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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

[F-O]Friedman, N. A. & Ornstein, D. S.. On isomorphism of weak Bernoulli transformations. 4 (1970), 337–352.Google Scholar
[F]Furstenberg, H.. Disjointness in ergodic theory, minimal sets and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1–49.CrossRefGoogle Scholar
[J]Johnson, A. S. A.. Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers. Preprint.Google Scholar
[K]Keane, M. S. & Pearce, C. E. M.. On normal numbers. J. Austr. Math. Soc. (series A) 32 (1982), 79–87.Google Scholar
[L]Lyons, R.. On measures simultaneously 2- and 3-invariant. Israel Math. J. 61 (1988), 219–224.CrossRefGoogle Scholar
[R]Rudolph, D.. ×2 and ×3 invariant measures and entropy. Ergod. Th. & Dynam. Sys. 10 (1990), 395–406.CrossRefGoogle Scholar
[S]Schmidt, W.. On normal numbers. Pacific J. Math. 10 (1960), 661–672.CrossRefGoogle Scholar
[W]Weyl, H.. Über di Gleichverteilung von Zahlen mod Eins. Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar