Fourier-Stieltjes Coefficients and Asymptotic Distribution Modulo 1

1985 ◽  
Vol 122 (1) ◽  
pp. 155 ◽  
Author(s):  
Russell Lyons
Keyword(s):  
Asymptotic Distribution ◽  
Distribution Modulo 1 ◽  
Distribution Modulo

scholarly journals Mixing and asymptotic distribution modulo 1

1988 ◽  
Vol 8 (4) ◽  
pp. 597-619 ◽  
Author(s):  
Russell Lyons
Keyword(s):  
Asymptotic Distribution ◽  
Asymptotic Distributions ◽  
Strong Mixing ◽  
Arbitrary Sequence ◽  
Mixing Conditions ◽  
Distribution Modulo 1 ◽  
Stable Ergodicity ◽  
Bernoulli Convolutions ◽  
Riesz Products ◽  
Distribution Modulo

AbstractIf μ is a probability measure which is invariant and ergodic with respect to the transformationx↦qxon the circle ℝ/ℤ, then according to the ergodic theorem, {qnx} has the asymptotic distribution μ for μ-a.e.x. On the other hand, Weyl showed that when μ is Lebesgue measure, λ, and {mj} is an arbitrary sequence of integers increasing strictly to ∞, the asymptotic distribution of {mjx} is λ for λ-a.e.x. Here, we investigate the asymptotic distributions of {mjx} μ-a.e. for fairly arbitrary {mj} under some strong mixing conditions on μ. The result is a kind of stable ergodicity: the distributions are obtained from simple operations applied to μ. The ideas extend to the situation of a sequence of transformationsx↦qnxwhere invariance is not present. This gives us information about many Riesz products and Bernoulli convolutions. Finally, we apply the theory to resolve some questions aboutH-sets.

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