Structure of the distribution of one class of generalized Bernoullie convolutions

According to the Jessen-Wintner theorem, the sum of a convergent with probability unity random series with independent discretely distributed terms has a pure Lebesgue type of distribution, i.e. discrete, absolutely continuous, or singular. Levi's theorem gives necessary and sufficient conditions for the corresponding distribution (Jessen-Wintner type distribution) to be discrete. In the general case, the problem of finding criteria for absolute continuity (singularity) of Jessen-Wintner type distributions is complex, therefore it is considered by researchers in certain classes. An important class of distributions that satisfy the Jessen-Wintner theorem are classical Bernoulli convolutions and their corresponding generalizations (generalized Bernoulli convolutions), which have been the subject of research by many scientists. Interest in the corresponding problem is still high. Special difficulties in the study of the Lebesgue structure of Jessen-Wintner type distributions, in particular in the class of generalized Bernoulli convolutions, arise for the case of generalized Bernoulli convolutions with substantial overlaps. For generalized Bernoulli convolutions with substantial overlaps, classical approaches of measure theory, based in particular on the Kakutani theorem, in many cases allow obtaining only certain sufficient conditions for the absolute continuity (singularity) of the corresponding distribution function. In addition to the classical problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions, the following are relevant: the problem of studying the topological-metric structure of the distribution spectrum, fractal analysis of the distribution supports, asymptotic properties of the characteristic function at infinity, and others. The paper specifies necessary and sufficient conditions for the absolute continuity and singularity of the distribution of one class of generalized symmetric Bernoulli convolutions. The corresponding results are a generalization of the results with respect to the Lebesgue structure of the distribution of a random variable with independent s-digits. The emphasis is also placed on proving that the spectrum of the corresponding class of distributions considered in the paper has a positive Lebesgue measure.