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

TL;DR

This paper investigates the fractal and spectral properties of Bernoulli convolutions generated by the second Ostrogradsky series, revealing their singular, Cantor-like distribution with zero Hausdorff dimension and non-vanishing Fourier transform at infinity.

Contribution

It introduces new methods to analyze the irregularity and fractal structure of these convolutions, extending understanding of their spectral and dimensional properties.

Findings

The distribution is singular with zero Hausdorff dimension.

Fourier-Stieltjes transform does not tend to zero at infinity.

Conditions for Hausdorff--Billingsley dimension preservation are established.

Abstract

We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} \xi = \sum_{k=1}^\infty \frac{(-1)^{k+1}\xi_k}{q_k}, \end{equation} where $q_k$ is a sequence of positive integers with $q_{k+1}\geq q_k(q_k+1)$, and $\{\xi_k\}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $\xi$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_{\xi})=0$) whose Fourier-Stieltjes transform does not tend to zero at infinity. We also develop different approaches how to estimate a level of "irregularity" of probability distributions whose spectra are of zero Hausdorff dimension. Using generalizations of the Hausdorff measures and dimensions, fine fractal properties of the probability measure $\mu_\xi$ are studied in details. Conditions for the Hausdorff--Billingsley dimension preservation on the spectrum by its probability distribution function are also obtained.