Geodesic
On non-trivial additive cocycles on the torus
Sbornik. Mathematics, Tome 199 (2008) no. 2, pp. 229-251

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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 $$ w(x+\alpha)-w(x)=f(x) $$ has no measurable solutions $w$ for any badly approximable vector $\alpha$. For every vector $\alpha$ 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. 
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     author = {A. V. Rozhdestvenskii},
     title = {On non-trivial additive cocycles on the torus},
     journal = {Sbornik. Mathematics},
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     volume = {199},
     number = {2},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2008_199_2_a3/}
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A. V. Rozhdestvenskii. On non-trivial additive cocycles on the torus. Sbornik. Mathematics, Tome 199 (2008) no. 2, pp. 229-251. http://geodesic.mathdoc.fr/item/SM_2008_199_2_a3/