  It is a typical standard assumption in the density deconvolution problem that
the characteristic function of the measurement error distribution is non-zero
on the real line. While this condition is assumed in the majority of existing
works on the topic, there are many problem instances of interest where it is
violated. In this paper we focus on non--standard settings where the
characteristic function of the measurement errors has zeros, and study how
zeros multiplicity affects the estimation accuracy. For a prototypical problem
of this type we demonstrate that the best achievable estimation accuracy is
determined by the multiplicity of zeros, the rate of decay of the error
characteristic function, as well as by the smoothness and the tail behavior of
the estimated density. We derive lower bounds on the minimax risk and develop
optimal in the minimax sense estimators. In addition, we consider the problem
of adaptive estimation and propose a data-driven estimator that automatically
adapts to unknown smoothness and tail behavior of the density to be estimated.
