I will define a new variant of the Seshadri constant for ample line bundles in positive characteristic. We will then explore how lower bounds for this constant imply the global generation and/or very ampleness of the corresponding adjoint line bundle. As a consequence, we will deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic (even though we may lack the usual vanishing theorems). This is joint work with Mircea Mustata.