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In this article, we prove that from any sequence of balls whose associatedlimsup set has full $\mu$-measure, one can extract a well-distributedsubsequence of balls. From this, we deduce the optimality of various lowerbounds for the Hausdorff dimension of limsup sets of balls obtained by masstransference principles. We also establish a version of Borel-Cantellidivergence lemma particulary suited for limsup set generated by balls. Thislemma is very similar to the one proved by Bersnevich and Velani but themeasure is not assumed ot be doubling.

Extraction of optimal subsequences of sequence of balls, and application to optimality estimates of mass transference principles