  We consider probability measures on the real line or unit circle with Jacobi
or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized
bounded variation condition. This latter condition requires that a sequence can
be expressed as a sum of sequences $\beta^{(l)}$, each of which has rotated
bounded variation, i.e., $\sum_{n=0}^\infty | e^{i\phi_l} \beta_{n+1}^{(l)} -
\beta_n^{(l)} |$ is finite for some $\phi_l$. This includes discrete
Schr\"odinger operators on a half-line or line with finite linear combinations
of Wigner--von Neumann type potentials.
  For the real line, we prove that in the Lebesgue decomposition $d\mu=f dm +
d\mu_s$ of such measures, the intersection of (-2,2) with the support of
$d\mu_s$ is contained in an explicit finite set S (thus, $d\mu$ has no singular
continuous part), and f is continuous and non-vanishing on $(-2,2) \setminus
S$. The results for the unit circle are analogous, with (-2,2) replaced by the
unit circle.
