Related papers: Numbers with simply normal $\beta$-expansions

Let $X$ be a Hadamard manifold with pinched negative curvature $-b^2\leq\kappa\leq -1$. Suppose $\Sigma\subseteq X$ is a totally geodesic, codimension-1 submanifold and consider the geodesic flow $\Phi^\nu_t$ on $X$ generated by a unit…

Given $\beta>1$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$, defined by $T_\beta(x)=\beta x-\lfloor \beta x\rfloor$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose…

The coefficients $A_n(\alpha,\beta,\omega)$ in the Maclaurin expansion $(1+\omega z)^{\alpha}(1-z)^{-\beta}= \sum_{n=0}^{\infty} A_n(\alpha,\beta,\omega)z^n$ are studied, where $\omega,z \in \mathbb{C}$ with $|z| < |\omega|=1$, and…

For a fixed $\theta^2=1/m$, $m \in \mathbb{N}_+$, let $x \in [0, \theta)$ and $[a_1(x) \theta, a_2(x) \theta, \ldots]$ be the $\theta$-expansion of $x$. Our first goal is to extend for $\theta$-expansions the results of Jarnik \cite{J-1928}…

We obtain a Gessel-type expansion in Jack polynomials for the expectations of multiplicative functionals in the circular $\beta$-ensemble. As a consequence, we establish a Szeg\H{o}-type limit theorem for all $H^{1/2}(\mathbb{T})$ functions…

We consider an analogue of Nakada's $\alpha$-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given $\alpha \in [\frac{1}{2}(\sqrt{5}-1),\frac{1}{2}(\sqrt{5}+1)]$, we show…

We prove the existence of a 1/N expansion to all orders in beta matrix models with a confining, off-critical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained…

Fix a sequence of integers $Q=\{q_n\}_{n=1}^\infty$ such that $q_n$ is greater than or equal to 2 for all $n$. In this paper, we improve upon results by J. Galambos and F. Schweiger showing that almost every (in the sense of Lebesgue…

We generalize the greedy and lazy $\beta$-transformations for a real base $\beta$ to the setting of alternate bases $\boldsymbol{\beta}=(\beta_0,\ldots,\beta_{p-1})$, which were recently introduced by the first and second authors as a…

Given a positive integer $M$ and a real number $q >1$, a \emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $(c_i) \in \{0,\ldots,M\}^\infty$ such that \[x=\sum_{i=1}^{\infty} c_iq^{-i}.\] It is well known…

In this paper, we consider the following singular integral \begin{equation*} T_jf(x)=K_j*f(x), K_j(x)=\frac{x_j}{|x|^{n+1-\beta}}, \end{equation*} where $x\in R^n, 0\le \beta<n, j=1,2,\cdots, n$. When $\beta=0$, it corresponds to the Riesz…

Let $X=\sum_{k=1}^\infty X_k \beta^{-k}$ be the base-$\beta$ expansion of a continuous random variable $X$ on the unit interval where $\beta$ is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled…

We compare the level zero part of the type of a representation of GL(n) over a non-archimedean local field with the tame part of its Langlands parameter restricted to inertia. By normalizing this comparison, we construct canonical…

S. Baker (2019), B. B\'ar\'any and A. K\"{a}enm\"{a}ki (2019) independently showed that there exist iterated function systems without exact overlaps and there are super-exponentially close cylinders at all small levels. We adapt the method…

A unified proof of the irrationality of the special values L(n, X), n > 1 an integer, of the beta L-function is put forward in this note. The first case of n = 2 seems to confirm that the Catalan constant L(2, X) is an irrational number.

Let $\alpha, \beta$ be two relatively prime algebraic integers in a number field $K$ and $N$ be a positive integer. We show that the number of $n\in\{1,2,\dots,N\}$ such that the $\beta$-adic expansion of $\alpha^n$ omits a given digit is…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

We prove that at least one of the six numbers $\beta(2i)$ for $i=1,\dots,6$ is irrational. Here $\beta(s)=\sum_{k=0}^\infty(-1)^k(2k+1)^{-s}$ denotes Dirichlet's beta function, so that $\beta(2)$ is Catalan's constant.

We introduce an ergodic approach to the study of {\em joint normality} of representations of numbers. For example, we show that for any integer $b \geq 2$ almost every number $x \in [0,1)$ is jointly normal with respect to the $b$-expansion…

We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama's result for positive Pisot unit base $\beta$, we find a sufficient condition so that there exist an interval $J$…