On Bernoulli convolutions generated by second Ostrogradsky series and their fine fractal properties

Abstract We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables ξ = ∑ k = 1 ∞ ( - 1 ) k + 1 ⁢ ξ k q k , see text \xi=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}\xi_{k}}{q_{k}}, where q k {q_{k}} is a sequence of positive integers with q k + 1 ≥ q k ⁢ ( q k + 1 ) {q_{k+1}\geq q_{k}(q_{k}+1)} , and { ξ k } {\{\xi_{k}\}} are independent random variables taking the values 0 and 1 with probabilities p 0 ⁢ k {p_{0k}} and p 1 ⁢ k {p_{1k}} respectively. We prove that ξ has an anomalously fractal Cantor-type singular distribution ( dim H ⁡ ( S ξ ) = 0 {\dim_{H}(S_{\xi})=0}