A mass transference principle and sets with large intersections

Abstract: I prove a mass transference principle for general shapes, similar to a recent result by H. Koivusalo and M. Rams. The proof relies on Vitali's covering lemma and manipulations with Riesz energies. The main novelty is that it is proved that the obtained limsup-set belongs to the classes of sets with large intersections, as defined by K. Falconer. This has previously not been proved for as general shapes as in this paper.