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While much work has been done to characterize the Turing degrees computingmembers of various collections of fast growing functions, much less has beendone to characterize the rate of growth necessary to compute particulardegrees. Prior work has shown that every degree computed by all sufficientlyfast growing functions is uniformly computed by all sufficiently fast growingfunctions. We show that the rate of growth sufficient for a function touniformly compute a given Turing degree can be separated by an arbitrary numberof jumps from the rate of growth that suffices for a function to non-uniformlycompute the degree. These results use the unpublished method Harringtondeveloped to answer McLaughlin's conjecture so we begin the paper with arigorous presentation of the approach Harrington sketched in his handwrittennotes on the conjecture. We also provide proofs for the important computabilitytheoretic results Harrington noted were corollaries of this approach. Inparticular we provide the first published proof of Harrington's result thatthere is an effectively given sequence of {\Pi}^0_1 singletons that areLow_{\alpha} none of which is computable in the effective join of the {\alpha}jumps of the others for every computable ordinal {\alpha}.

Harrington's Solution to McLaughlin's Conjecture and Non-uniform Self-moduli