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In this note we prove an abstract version of a recent quantitativestratifcation priciple introduced by Cheeger and Naber (Invent. Math., 191(2013), no. 2, 321-339; Comm. Pure Appl. Math., 66 (2013), no. 6, 965-990).Using this general regularity result paired with an $\varepsilon$-regularitytheorem we provide a new estimate of the Minkowski dimension of the set ofhigher multiplicity points of a Dir-minimizing Q-valued function. The abstractpriciple is applicable to several other problems: we recover recent results inthe literature and we obtain also some improvements in more classical contexts.

Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result