Continued fractions and irrational rotations

Abstract

Let 0<α<1 be an irrational number and s be a real number. If the irrational number number α has bounded partial quotients, the function $exp(2\pi is \chi_{[o,t)})$ on the unit circle is a constant multiple of coboundary if and only if t is an integer multiple of α. However, if the irrational α has unbounded partial quotients, it is unknown for which t $exp(2\pi is \chi_{[o,t)})$ is a constant multiple of coboundary. Let both $r_1$ and $r_2$ are rational numbers and α be any irrational number. In this paper, it is shown that for any real s, the function $exp(2\pi is\chi_{[0,r_1\alpha+r_2)}$ is a constant multiple of a cobaundary if and only if $r_1,r_2$ are integer.