Constructing self-similar subsets within the fractal support of Lacunary Wavelet Series for their multifractal analysis

Given a fractal $\mathcal{I}$ whose Hausdorff dimension matches with the
upper-box dimension, we propose a new method which consists in selecting inside
$\mathcal{I}$ some subsets (called quasi-Cantor sets) of almost same dimension
and with controled properties of self-similarties at prescribed scales. It
allows us to estimate below the Hausdorff dimension $\mathcal{I}$ intersected
to limsup sets of contracted balls selected according a Bernoulli law, in
contexts where classical Mass Transference Principles cannot be applied. We
apply this result to the computation of the increasing multifractal spectrum of
lacunary wavelet series supported on $\mathcal{I}$.