  In an earlier paper (joint with Min Ru), we proved a result on diophantine
approximation to Cartier divisors, extending a 2011 result of P. Autissier.
This was recently extended to certain closed subschemes (in place of divisors)
by Ru and Wang. In this paper we extend this result to a broader class of
closed subschemes. We also show that some notions of $\beta(\mathscr L,D)$
coincide, and that they can all be evaluated as limits.
