Superfractality of the set of incomplete sums of one positive series

Abstract

We consider a family of convergent positive normed series with real terms defined by the conditions $$\sum ^{\infty}_{n=1} d_n = \underbrace{c_1 + ...+c_1}_{a_1} + \underbrace{c_2 + ...+c_2}_{a_2} + ... + \underbrace{c_n + ...+c_n}_{a_n} + \widetilde{ r_n} = 1,$$ where $(a_n)$ is a nondecreasing sequence of real numbers. The structural properties of these series are investigated. For a partial case, namely, $(a_n) = 2^{n - 1}, c_n = (n + 1)\widetilde {r_n}, n \in N$, we study the geometry of the series (i.e., the properties of cylindrical sets, metric relations generated by them, and topological and metric properties of the set of all incomplete sums of the series). For the infinite Bernoulli convolution determined we describe its Lebesgue structure (discrete, absolutely continuous, and singular components) and spectral properties, as well as the behavior of the absolute value of the characteristic function at infinity. We also study the finite autoconvolutions of distributions of this kind.