On the effect of projections on the Billingsley dimensions

Abstract

We are interested in the behavior of Billingsley dimensions under projections onto a lower dimensional linear subspace. The results in this paper establish the connections with various dimensions of subsets [Formula: see text] of [Formula: see text] and their projections, and generalize many known results about Hausdorff and packing dimensions of projections of [Formula: see text]. In particular, we improve, through these results, one of the main theorems of Selmi et al. in [Multifractal variation for projections of measures, Chaos Solitons Fractals 91 (2016) 414–420] and treat an unsolved case in their work.