We study conditions that will ensure that a crossed product of a C-algebra by a
discrete exact group is purely infinite (simple or non-simple). We are particularly interested
in the case of a discrete non-amenable exact group acting on a commutative C-algebra,
where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the
spectrum of the abelian C-algebra. As an application of our results we show that every
discrete countable non-amenable exact group admits a free amenable minimal action on the
Cantor set such that the corresponding crossed product C-algebra is a Kirchberg algebra
in the UCT class.