Related papers: The $\beta$-transformation with a hole at 0

For the two-parameter Mittag-Leffler function $E_{\alpha,\beta}$ with $\alpha > 0$ and $\beta \ge 0,$ we consider the question whether $|E_{\alpha,\beta}(z)|$ and $E_{\alpha,\beta}(\Re z)$ are comparable on the whole complex plane. We show…

Let $(\Sigma, \sigma)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\in\Sigma$ to the $n$-th cylinder containing $x$. Denote $$E^\varphi_{\alpha,\beta}=\left\{x\in\Sigma:…

Let $I=[0,1)$, $b\in \{2,3,\ldots\}$ and $f:I\to I$ be an injective piecewise $\frac{1}{b}$-affine map, that is, assume that there exists a partition of $I$ into intervals $I_1,\ldots,I_n$ such that $\vert f(x)-f(y)\vert\le\frac1b \vert…

Let Q_i, i=1,...,t, be real nondegenerate indefinite quadratic forms in d variables. We investigate under what conditions the closure of the set {(Q_1(x),...,Q_t(x)): x\in Z^d-{0}} contains (0,..,0). As a corollary, we deduce several…

We study the superconducting properties of a two-dimensional superconductor in the proximity to an electronic topological transition (ETT). In contrast to the 3D case, we find that the superconducting gap at T=0, the critical temperature…

We study an interval exchange transformation of [0,1] formed by cutting the interval at the points 1/n and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero.…

If E(z,s) is the nonholomorphic Eisenstein series on the upper half plane, then for all y sufficiently large, E(z,s) has a "Siegel zero." That is E(z,\beta)=0 for a real number \beta just to the left of one. We give a generalization of this…

It is a longstanding conjecture that given a subset $E$ of a metric space, if $E$ has finite Hausdorff measure in dimension $\alpha\ge 0$ and $\mathscr{H}^\alpha\llcorner E$ has unit density almost everywhere, then $E$ is an…

In this paper, we demonstrate that the modified Emden type equation (MEE), $\ddot{x}+\alpha x\dot{x}+\beta x^3=0$, is integrable either explicitly or by quadrature for any value of $\alpha$ and $\beta$. We also prove that the MEE possesses…

Let $\psi : \mathbb{R}_{>0}\rightarrow \mathbb{R}_{>0}$ be a non-increasing function. Denote by $W(\psi)$ the set of $\psi$-well-approximable points and by $E(\psi)$ the set of points $x\in[0,1]$ such that for any $0 < \epsilon < 1$ there…

Let $\beta > 1$ be a real number and $(\epsilon_1(x, \beta), \epsilon_2(x, \beta), \ldots)$ be the $\beta$-expansion of a point $x \in (0, 1]$. For all $x \in (0,1]$, let $A(D(x))$ be the set of accumulation points of $\frac{-\log_\beta…

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different…

The finite Hilbert transform $T$ is a classical (singular) kernel operator which is continuous in every rearrangement invariant space $X$ over $(-1,1)$ having non-trivial Boyd indices. For $X=L^p$, $1<p<\infty$, this operator has been…

Let H be a separable Hilbert space with a fixed orthonormal basis (e_n), n>=1, and B(H) be the full von Neumann algebra of the bounded linear operators T: H -> H. Identifying l^\infty = C(\beta N) with the diagonal operators, we consider…

Fix an alphabet $A=\{0,1,\dots,M\}$ with $M\in\mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in(1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but…

For $\lambda>0$, let $E_{\lambda}$ be the self-similar set generated by the iterated function system (IFS) $\left \{ \frac{x}{3}, \frac{x+\lambda}{3} \right \}$. In this paper we study the structure of parameters $\lambda$ in which…

Thermodynamics in the vicinity of a critical endpoint with nonclassical exponents $\alpha$, $\beta$, $\gamma$, $\delta$, $...$ is analyzed in terms of density variables (mole fractions, magnetizations, etc.). The shapes of the isothermal…

We consider the Helmholtz equation with real analytic coefficients on a bounded domain $\Omega\subset\mathbb{R}^{d}$. We take $d+1$ prescribed boundary conditions $f^{i}$ and frequencies $\omega$ in a fixed interval $[a,b]$. We consider a…

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove…

This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called {\em $\beta$-expansions}) and the infinite derivatives of a class of self-affine functions. Precisely, we extend Okamoto's…