Related papers: Bernoulli convolutions -- 2023

We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by $$ S=\sum_{n=1}^{\infty}\frac{I_n}{n^s}, $$ where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value $1$ with…

If $S$ is an infinite sequence over a finite alphabet $\Sigma$ and $\beta$ is a probability measure on $\Sigma$, then the {\it dimension} of $ S$ with respect to $\beta$, written $\dim^\beta(S)$, is a constructive version of Billingsley…

Let $\xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(\xi)$ denote an $n\times n$ random matrix with entries that are independent copies of $\xi$. For $\xi$ which is not uniform on its support, we show…

Let $(B_{i})$ be a sequence of measurable sets in a probability space $(X,\mathcal{B}, \mu)$ such that $\sum_{n=1}^{\infty} \mu (B_{i}) = \infty$. The classical Borel-Cantelli lemma states that if the sets $B_{i}$ are independent, then $\mu…

We study probability measures defined by the variation of the sum of digits in the Zeckendorf representation. For $r\ge 0$ and $d\in\mathbb{Z}$, we consider $\mu^{(r)}(d)$ the density of integers $n\in\mathbb{N}$ for which the sum of digits…

Early results by Borel and Cantelli and Erd\H{o}s and Chung have provided bounds for the measure of a limsup set in terms of measures of its constituent sets and their intersections. Recent work by Beresnevich and Velani \cite{Velanipaper}…

This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients…

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and…

Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a…

We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble…

We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit…

We give an explicit expression for the invariant measure, absolutely continuous with respect to the Lebesgue measure, of the greedy beta-transformation with three deleted digits. We define a version of the natural extension of the…

Consider a random $n\times n$ zero-one matrix with "density" $p$, sampled according to one of the following two models: either every entry is independently taken to be one with probability $p$ (the "Bernoulli" model), or each row is…

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events on a probability space $(\Omega,\mathcal{F},\mathbf{P})$. We show that if $\lim_{m\to\infty}\sum_{n=1}^{m}w_n\mathbf{P}(A_n)=\infty$ where each $w_n\in\mathbb{R}$, then…

We investigate variance bounds under symmetry constraints in classical, free, and Boolean probability, focusing on Bernoulli distributions and their noncommutative analogues, projections with trace \(p\). We show that symmetrizers under…

The Menon-Sury's identity is as follows: \begin{equation*} \sum_{\substack{1 \leq a, b_1, b_2, \ldots, b_r \leq n\\\mathrm{gcd}(a,n)=1}} \mathrm{gcd}(a-1,b_1, b_2, \ldots, b_r,n)=\varphi(n) \sigma_r(n), \end{equation*} where $\varphi$ is…

Consider a random sum $\eta_1 v_1 + ... + \eta_n v_n$, where $\eta_1,...,\eta_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $\P(\eta_1 v_1 + ...…

We explore an asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.

We present a new formula for the Bernoulli numbers as the following integral $$B_{2m} =\frac{(-1)^{m-1}}{2^{2m+1}} \int_{-\infty}^{+\infty} (\frac{d^{m-1}}{dx^{m-1}} {sech}^2 x)^2dx. $$ This formula is motivated by the results of Fairlie…

We study the ergodic properties of a class of measures on $\Sigma^{\mathbb{Z}}$ for which $\mu_{\mathcal{A},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\left \|A_{x_{0}}\cdots A_{x_{n-1}}\right \| ^{t}$, where $\mathcal{A}=(A_{0}, \ldots ,…