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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.

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