  Using a perturbative argument, we show that in any finite region containing
the lowest transverse eigenmode, the spectrum of a periodically curved smooth
Dirichlet tube in two or three dimensions is absolutely continuous provided the
tube is sufficiently thin. In a similar way we demonstrate absolute continuity
at the bottom of the spectrum for generalized Schr\"odinger operators with a
sufficiently strongly attractive $\delta$ interaction supported by a periodic
curve in $\mathbb{R}^d, d=2,3$.
