On non-trivial additive cocycles on the torus

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.
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