On the ergodicity of Weyl sum cocycles
2007 ◽
Vol 27
(6)
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pp. 1851-1863
Keyword(s):
AbstractWe present the quadratic Weyl sums $\sum _{k=0}^{n-1} e^{2\pi i(k^2\theta +2kx)}$ with θ,x∈[0,1) as cocycles over a measure-preserving transformation on the two-dimensional torus. We show then that these cocycles are not coboundaries for every irrational θ∈[0,1), and that for a dense Gδ set of θ∈[0,1) the corresponding skew product is ergodic. For each of those θ, there exists a dense Gδ set of full measure of x∈[0,1) for which the sequence $\sum _{k=0}^{n-1} e^{2\pi i(k^2\theta +2kx)}$, n=1,2,… , is dense in $\mathbb {C}$.
Keyword(s):
Keyword(s):
2017 ◽
Vol 39
(3)
◽
pp. 764-794
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2011 ◽
Vol 75
(5)
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pp. 1007-1045
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Keyword(s):
1981 ◽
Vol 1981
(328)
◽
pp. 1-8
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1998 ◽
Vol 115
(1)
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pp. 448-457
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