A fundamental property connecting the symbolic powers and
the usual powers of ideals in regular rings was discovered by
Ein, Lazarsfeld, and Smith in 2001, and later extended by
Hochster and Huneke in 2002. In this paper we give further
generalizations which give better results in case the quotient
of the regular ring by the ideal is F-pure or F-pure type. Our
methods also give insight into a conjecture of Eisenbud and
Mazur concerning the existence of evolutions. The methods used
come from tight closure and reduction to positive
characteristic.