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On dichotomy of Riesz products

Published online by Cambridge University Press:  24 October 2008

G. Ritter
G. Ritter
Affiliation:
(Mathematisches Institut, Erlangen, Federal Republic of Germany and University of Washington, Seattle)
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Extract

Background. Riesz products are very useful for the construction of singular measures on compact, Abelian groups. Under some circumstances, two Riesz products are either equivalent or singular in the measure-theoretic sense. Exact knowledge of these circumstances has been of major interest ever since the 1930s, when Riesz's famous example (8) was recognized as a fertile source of examples of singular continuous measures. Zygmund(11) showed that any Riesz product over a Hadamard dissociate subset of ℕ is either a square integrable function or singular with respect to Lebesgue measure. Hewitt–Zuckerman(4) generalized these products to all compact, Abelian groups, introducing the notion of a dissociate subset. They extended Zygmund's result in certain cases. The next major step was taken by Brown–Moran(3) and Peyrière(6), (7), who showed that two Riesz products

are mutually singular if

The author (9) has improved another result of Brown–Moran (3) by showing that µa and µb are equivalent if

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 1 , January 1979 , pp. 79 - 89
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

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