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This paper establishes uniform bounds in characteristic $p$ rings which areeither F-finite or essentially of finite type over an excellent local ring.These uniform bounds are then used to show that the Hilbert-Kunz lengthfunctions and the normalized Frobenius splitting numbers defined on theSpectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunzmultiplicity function and the F-signature function. From this we establish thatthe F-signature function is lower semi-continuous. We also give a new proof ofthe upper semi-continuity of Hilbert-Kunz multiplicity, which was originallyproven by Ilya Smirnov.

Uniform Bounds in F-Finite Rings and Lower Semi-Continuity of the F-Signature