Local Okounkov bodies and limits in prime characteristic
  This article is concerned with the asymptotic behavior of certain sequences
of ideals in rings of prime characteristic. These sequences, which we call
$p$-families of ideals, are ubiquitous in prime characteristic commutative
algebra (e.g., they occur naturally in the theories of tight closure,
Hilbert-Kunz multiplicity, and $F$-signature). We associate to each $p$-family
of ideals an object in Euclidean space that is analogous to the Newton-Okounkov
body of a graded family of ideals, which we call a $p$-body. Generalizing the
methods used to establish volume formulas for the Hilbert-Kunz multiplicity and
$F$-signature of semigroup rings, we relate the volume of a $p$-body to a
certain asymptotic invariant determined by the corresponding $p$-family of
ideals. We apply these methods to obtain new existence results for limits in
positive characteristic, an analogue of the Brunn-Minkowski theorem for
Hilbert-Kunz multiplicity, and a uniformity result concerning the positivity of
a $p$-family.
