An Additive Cohomological Equation and Typical Behavior of Birkhoff Sums over a Translation of the Multidimensional Torus

For a periodic function $f$ with a given decrease of the moduli of its Fourier coefficients, we analyze the solvability of the equation $w(T_\alpha x)-w(x)=f(x)-\int_{\mathbb T^d}f(t)\,dt$ and the asymptotic behavior of the Birkhoff sums $\sum _{s=0}^{n-1} f(T^s_\alpha x)$ for almost every $\alpha$. The results obtained are applied to the study of ergodic properties of a cylindrical cascade and of a special flow on the torus.