The talk is devoted to the study of fractal properties of infinite symmetric Bernoulli convolutions with essential overlaps generated by positive convergent series with additional regularity conditions. Some special subclasses of such distributions were studied by Ya. Gontcharenko, M. Pratsiovytyi, G.Torbin and others.

In our talk we shall demonstrate several new probabilistic and dimensional phenomena related to such type Bernoulli convolutions. In particular, we show that the Gontcharenko-Pratsiovytyi-Torbin conjecture on the Hausdorff-Besicovitch dimension of spectra of such probability distributions is not true.

The obtained results stress the importance of investigations devoted to  problems of faithfulness resp. non-faithfulness of Vitaly coverings.