  We prove an analogue of the L\"uck Approximation Theorem in positive
characteristic for certain residually finite rationally soluble (RFRS) groups
including right-angled Artin groups and Bestvina--Brady groups. Specifically,
we prove that the mod $p$ homology growth equals the dimension of the group
homology with coefficients in a certain universal division ring and this is
independent of the choice of residual chain. For general RFRS groups we obtain
an inequality between the invariants. We also consider a number of applications
to fibring, amenable category, and minimal volume entropy.
