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

Uniform bounds in F-finite rings and lower semi-continuity of the F-signature