We study reparameterizations of irrational linear flows on the torus, or equivalently special flows over irrational circle rotations. If the rotation in the base is Diophantine and the ceiling function is analytic then the special flow is analytically conjugate to an irrational linear flow. For any analytic ceiling function which is not a trigonometric polynomial there is an irrational rotation so that the special flow over this rotation and under the ceiling function is weak mixing. More exotic behaviours are also possible. In joint work with B. Fayad and A. Katok we showed that for any Liouville rotation there exists a smooth ceiling function for which the special flow has a mixed spectrum. For certain rotations we obtain analytic ceiling functions. Due to the special nature of these constructions A. Katok conjectured that for functions with regular decay of their Fourier coefficients mixed spectra might not be possible. Together with B. Fayad we prove that indeed for functions with sufficiently regular exponential decay of Fourier coefficients there exists a dichotomy: depending on the irrational rotation the special flow is either weak mixing or L^2 conjugate to an irrational linear flow (and hence has discrete spectrum). Strangely the conditions we arrive at depend on conditions on Fourier coefficients along arithmetic progressions and apply to some functions with decidedly unregular decay.