Birational Nevanlinna constants, beta constants, and diophantine approximation to closed subschemes

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.