Superfractality of the set of incomplete sums of one positive series (Q2274575)

The problem of topological and metric properties of the set of incomplete sums of an absolutely convergent series was formulated by \textit{S.~Kakeya} [Sci. Rep. Tôhoku Univ., I. Ser.~3, 159--163 (1915; JFM 45.0377.06)]. By following this direction and by considering a family of convergent positive normed series with real terms by the conditions \[ \sum^\infty_{n=1} d_n= c_1+\cdots+ c_1+\cdots+ c_n+\cdots+ c_n+ {\tilde r}_n=1, \] (\(c_n\) appears \(\alpha_n\) times, \(n= 1,2,3,\dots\)), where \((\alpha_n)_n\) is a nondecreasing sequence of natural numbers, the authors manage to investigate the structural properties of these series. Moreover, by restricting to the case where \(\alpha_n=2^{n-1}\) and \(c_n=(n+1){\tilde r}_n\), \(n=1,2,3,\dots\), the authors manage to study the geometry of the series. Regarding the infinite Bernoulli convolution specified by this series the authors manage to describe its Lebesgue structure and spectral properties and to study the behaviour of the absolute value of the characteristic function at infinity as well as finite convolutions of distributions of this kind.