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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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