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On non-trivial additive cocycles on the torus. (English. Russian original) Zbl 1170.37005
Sb. Math. 199, No. 2, 229-251 (2008); translation from Mat. Sb. 199, No. 2, 71-92 (2008).
Let $$T_\alpha: \mathbb{T}^d \to \mathbb{T}^d, T_\alpha x_i = x_i + \alpha_i\pmod 1$$, $$i = 1,\dots,d,$$ be the translation of the torus $$\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$$ by a fixed irrational vector $$\alpha = (\alpha_1,\dots,\alpha_n)$$ so that the numbers $$1, \alpha_1,\dots,\alpha_n$$ are linearly independent over $$\mathbb{Z}.$$ A function $$f: \mathbb{T}^d \to \mathbb{R}$$ is called a non-trivial additive cocycle over the transformation $$T_\alpha$$ if the equation $$w(T_\alpha x) - w(x) = f(x)$$ has no measurable solution $$w: \mathbb{T}^d \to \mathbb{R}.$$
Explicit examples of non-trivial additive cocycles over the transformation $$T_\alpha$$ for a badly approximable vector $$\alpha$$ are constructed. These cocycles are independent of $$\alpha$$ and possess the best smoothness in a certain sense.
For each vector $$\alpha$$ admitting an arbitrary prescribed degree of simultaneous Diophantine approximation, the author constructs a cocycle that is asymptotically normal in the strong sense and has an extreme smoothness.

##### MSC:
 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 11K60 Diophantine approximation in probabilistic number theory
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