Abstract:
We construct a family of functions f with zero mean on a multidimensional torus possessing a very
high degree of smoothness, such that the equation
w(x+α)−w(x)=f(x)
has no measurable solutions w for any badly approximable vector α. For every vector α
admitting an arbitrary prescribed degree of simultaneous Diophantine approximation we construct a cocycle of extremal smoothness that is asymptotically normal in the strong sense.
Bibliography: 19 titles.
\Bibitem{Roz08}
\by A.~V.~Rozhdestvenskii
\paper On non-trivial additive cocycles on the torus
\jour Sb. Math.
\yr 2008
\vol 199
\issue 2
\pages 229--251
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Linking options:
https://www.mathnet.ru/eng/sm3663
https://doi.org/10.1070/SM2008v199n02ABEH003917
https://www.mathnet.ru/eng/sm/v199/i2/p71
This publication is cited in the following 2 articles:
Cohen G., Lin M., “Joint and Double Coboundaries of Commuting Contractions”, Indiana Univ. Math. J., 70:4 (2021), 1355–1394
Goll M., Verbitskiy E., “Homoclinic Points of Principal Algebraic Actions”, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Lecture Notes in Applied Mathematics and Mechanics, 3, eds. Muntean A., Rademacher J., Zagaris A., Springer Int Publishing Ag, 2016, 251–292
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