Related papers: On the behavior of binary block-counting functions…

We show that the number of $1$'s in the first $N$ digits of the binary expansion of $\sqrt{2}$ is at least $\sqrt{2N}(1+o(1))$ and show that this bound can be improved to around $2\sqrt{N}/\sqrt{2\sqrt{2}-1}$ infinitely often.

We study Fourier-sparse Boolean functions over general finite Abelian groups. A Boolean function $f : G \to \{-1,+1\}$ is $s$-sparse if it has at most $s$ non-zero Fourier coefficients. We introduce a general notion of granularity of…

The Tu--Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base~$2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and $1\leq t<2^k-1$. Then \[\Bigl…

We have developed a heuristic showing that in the Dirichlet divisor problem for the almost all $n \in \mathbb{N}^{+}$: $$ R(n) \leq O(\psi(n)n^{\frac{1}{4}}) $$ where $$ R(n) = \Big\lvert \sum_{x=1}^{n}\Big\lfloor\frac{n}{x}\Big\rfloor -…

Understanding the multiplicity of stellar systems and the correlations between their hierarchical components provides crucial insights into star formation processes. If binary companions form independently in each component of a wide binary…

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same…

We show that the Hausdorff dimension of $(s,t)$-Furstenberg sets is at least $s+t/2+\epsilon$, where $\epsilon>0$ depends only on $s$ and $t$. This improves the previously best known bound for $2s<t\le 1+\epsilon(s,t)$, in particular…

Let $q$ be an integer $\geq 2$ and let $S_q(n)$ denote the sum of digits of $n$ in base $q$. For \[ \alpha=[0;\overline{1,m}],\ m\geq 2, \] let $S_{\alpha}(n)$ denote the sum of digits in the Ostrowski $\alpha$-representation of $n$. Let…

We investigate the distribution of the largest digit for a wide class of infinite parabolic Iterated Function Systems (IFSs) of the unit interval. Due to the recurrence to parabolic (neutral) fixed points, the dimension analysis of these…

We develop a fractional extension of the classical binomial distribution and the associated Bernstein operator, formulated within the framework of the generalized binomial theorem (Hara and Hino [Bull.\ London Math.\ Soc. \textbf{42}…

Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any…

We consider digital expansions to the base of $\tau$, where $\tau$ is an algebraic integer. For a $w \geq 2$, the set of admissible digits consists of 0 and one representative of every residue class modulo $\tau^w$ which is not divisible by…

Sidorov and Vershik showed that in base $G=\frac{\sqrt{5}+1}{2}$ and with the digits $0,1$ the numbers $x=nG ~(\text {mod} 1)$ have $\aleph_{0}$ expansions for any $n\in\mathbb{Z}$, while the other elements of $(0, \frac{1}{G-1})$ have…

The block counting process and the fixation line of the Bolthausen-Sznitman coalescent are analyzed. Spectral decompositions for their generators and transition probabilities are provided leading to explicit expressions for functionals such…

The Mittag-Leffler process $X=(X_t)_{t\ge 0}$ is introduced. This Markov process has the property that its marginal random variables $X_t$ are Mittag-Leffler distributed with parameter $e^{-t}$, $t\in [0,\infty)$, and the semigroup…

It has been recently shown that if $X$ is an $n\times N$ matrix whose entries are i.i.d. standard complex Gaussian and $l_1$ is the largest eigenvalue of $X^*X$, there exist sequences $m_{n,N}$ and $s_{n,N}$ such that…

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…

The aim of this paper is to investigate discrete approximations of the exponential functional $\int_0^{\infty} \exp(B(t) - \nu t) \di t$ of Brownian motion (which plays an important role in Asian options of financial mathematics) by the…

We construct extensions of the pure-jump $\Lambda$-Wright-Fisher processes with frequency-dependent selection ($\Lambda$-WF processes with selection) beyond their first passage time at the boundary $1$. We show that they satisfy some…

We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, obtaining analogues of the Siegel--Walfisz Theorem for large classes of multiplicative…