Related papers: The exceptional sets on the run-length function of…

We prove that the set of exceptional $\lambda\in (1/2,1)$ such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the…

Assuming the validity of Dickson's conjecture, we show that the set $\mathcal{V}$ of values of the Euler's totient function $\varphi$ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of…

Let $\beta_1,\beta_2>1$ and $T_i(x,y) = \bigl(\frac{x+i}{\beta_1}, \frac{y+i}{\beta_2}\bigr),\ i\in\{\pm1\}$. Let $A := A_{\beta_1, \beta_2}$ be the unique compact set satisfying $A = T_{1}(A) \cup T_{-1}(A)$. In this paper we give a…

An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ letters. If $u$ is a sequence with $r$ distinct letters, then let $\mathit{Ex}(u, n)$ be the maximum length of any $r$-sparse sequence with $n$ distinct letters which has…

If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta…

Two sets $A,B$ of positive integers are called \emph{exact additive complements}, if $A+B$ contains all sufficiently large integers and $A(x)B(x)/x\rightarrow1$. Let $A=\{a_1<a_2<\cdots\}$ be a set of positive integers. Denote $A(x)$ by the…

Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the…

Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$…

Let the sequence $\{t_n\}_{n=1}^{\infty}$ of reals satisfy the condition $ \frac{t_{n+1}}{t_n} \ge 1+ \frac{\gamma}{n^\beta},0\le \beta <1, \gamma >0. $ Then the set $ \{\alpha \in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||t_n…

Let $\{a_n\}_{n\in\mathbb{N}}$, $\{b_n\}_{n\in \mathbb{N}}$ be two infinite subsets of positive integers and $\psi:\mathbb{N}\to \mathbb{R}_{>0}$ be a positive function. We completely determine the Hausdorff dimensions of the set of all…

For $x\in (0,1)$, let $\langle d_1(x),d_2(x),d_3(x),\cdots \rangle$ be the Engel series expansion of $x$. Denote by $\lambda(x)$ the exponent of convergence of the sequence $\{d_n(x)\}$, namely \begin{equation*} \lambda(x)= \inf\left\{s…

The L\"uroth expansion of a real number $x\in (0,1]$ is the series \[ x= \frac{1}{d_1} + \frac{1}{d_1(d_1-1)d_2} + \frac{1}{d_1(d_1-1)d_2(d_2-1)d_3} + \cdots, \] with $d_j\in\mathbb{N}_{\geq 2}$ for all $j\in\mathbb{N}$. Given $m\in…

The derivative discontinuity in the exact exchange-correlation potential of ensemble Density Functional Theory (DFT) is investigated at the specific integer number that corresponds to the maximum number of bound electrons, $J_{max}$. A…

Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ be its continued fraction expansion with partial quotients ${a_n(x), n\ge 1}$. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to \infty$. In this…

We consider random Fibonacci sequences given by $x_{n+1}=\pm \beta x_{n}+x_{n-1}$. Viswanath (\cite{viswanath}), following Furstenberg (\cite{furst}) showed that when $\beta = 1$, $\lim_{n\to \infty}|x_{n}|^{1/n}=1.13...$, but his proof…

We study natural measures on sets of beta-expansions and on slices through self similar sets. In the setting of beta-expansions, these allow us to better understand the measure of maximal entropy for the random beta-transformation and to…

Let $\mathcal{B}$ be a collection of open sets in $\mathbb{R}^{n}$ such that, for any $x \in \mathbb{R}^{n}$, there exists a set $U \in \mathcal{B}$ of arbitrarily small diameter {containing $x$.} $\mathcal{B}$ is said to be a \emph{density…

Let $\omega=[a_1, a_2, \cdots]$ be the infinite expansion of continued fraction for an irrational number $\omega \in (0,1)$; let $R_n (\omega)$ (resp. $R_{n, \, k} (\omega)$, $R_{n, \, k+} (\omega)$) be the number of distinct partial…

By classical Fatou type theorems in various setups, it is well-known that positive harmonic functions have non-tangential limit at almost every point on the boundary. In this paper, in the setting of non-positively curved Harmonic manifolds…