  Hilbert-Kunz multiplicity and F-signature are numerical invariants of
commutative rings in positive characteristic that measure severity of
singularities: for a regular ring both invariants are equal to one and the
converse holds under mild assumptions. A natural question is for what singular
rings these invariants are closest to one. For Hilbert--Kunz multiplicity this
question was first considered by the last two authors and attracted significant
attention. In this paper, we study this question, i.e., an upper bound, for
F-signature and revisit lower bounds on Hilbert--Kunz multiplicity.
