Geodesic
Homological equations and topological properties of $S^1$-extensions over an ergodic rotation of the circle
Matematičeskie zametki, Tome 23 (1978) no. 3, pp. 463-470.

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A description is given of the set of $\beta\in[0;1]$, such that the homological equation $$ f(x+\beta)-f(x)=g(x+\alpha)-g(x) $$ has a continuous solution, where $f(x)$ is a continuous periodic function, $f(x+1)=f(x)$. The result obtained is applied in studying the property of relative separability of $S^1$-extensions over an ergodic rotation of the circle. 
@article{MZM_1978_23_3_a13,
     author = {A. A. Gura},
     title = {Homological equations and topological properties of $S^1$-extensions over an ergodic rotation of the circle},
     journal = {Matemati\v{c}eskie zametki},
     pages = {463--470},
     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_3_a13/}
}
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A. A. Gura. Homological equations and topological properties of $S^1$-extensions over an ergodic rotation of the circle. Matematičeskie zametki, Tome 23 (1978) no. 3, pp. 463-470. http://geodesic.mathdoc.fr/item/MZM_1978_23_3_a13/