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

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists…

Let $\beta>1$ be a real number. In this paper, the Hausdorff dimension of sets consisting of pairs of numbers with prescribed quantitative waiting time indicators in $\beta$-expansions are determined. More precisely, let $I$ be the unit…

Given $\beta\in(1,2)$, a $\beta$-expansion of a real $x$ is a power series in base $\beta$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of…

For a compact connected set $X\subseteq \ell^{\infty}$, we define a quantity $\beta'(x,r)$ that measures how close $X$ may be approximated in a ball $B(x,r)$ by a geodesic curve. We then show there is $c>0$ so that if $\beta'(x,r)>\beta>0$…

Consider $m \in \mathbb{N}$ and $\beta \in (1, m + 1]$. Assume that $a\in \mathbb{R}$ can be represented in base $\beta$ using a development in series $a = \sum^{\infty}_{n = 1}x(n)\beta^{-n}$ where the sequence $x = (x(n))_{n \in…

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on the complex projective space $\mathbb P^2$. For $\alpha > 2/5$ and $\beta = (2-2\alpha)/3$ we show that if $T$ has four point with Lelong number greater than…

Let $\alpha \in (0,2)$ and let $\beta>0$. Fix $-\pi<\varphi\leq \pi$ such that $|\varphi|>\alpha \pi/2$. We obtain asymptotic upper bounds on the Fourier transform of the radially symmetric tempered distribution \begin{equation*}…

In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $\beta$-expansions. More precisely, for an integer $d \ge 2$ and real numbers $\beta_i > 1$ $(1 \le i \le d)$, we consider the set of points…

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\dots\le|\lambda_d|$. We investigate the Hausdorff dimension of the…

In this paper, we prove that if $S\subseteq\mathbb{R}^d$ is hyperplane absolute winning on a closed hyperplane diffuse set $L\subseteq\mathbb{R}^d$, then $\mathrm{dim}_H S\cap K=\mathrm{dim}_H K$ for any irreducible self-conformal set…

We consider order preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(\ell_1, \ell_2)$. We detect a change in the geometry of the system. For $(\ell_1, \ell_2) \in [1,2]^2$ the geometry is…

We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $\beta$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain…

We introduce a parameter space containing all algebraic integers $\beta\in(1,2]$ that are not Pisot or Salem numbers, and a sequence of increasing piecewise continuous function on this parameter space which gives a lower bound for the…

We consider the Paneitz-type equation $\Delta^2 u -\alpha \Delta u +\beta (u-u^q ) =0$ on a closed Riemannian manifold $(M,g)$. We reduce the equation to a fourth-order ordinary differential equation assuming that $(M,g)$ admits a proper…

Let $X = \mathcal{A}^{\mathbb{Z}^d}$, where $d \geq 1$ and $\mathcal{A}$ is a finite set, equipped with the action of the shift map. For a given continuous potential $\phi: \mathcal{A}^{\mathbb{Z}^d} \to \mathbb{R}$ and $\beta>0$ (``inverse…

We show that given $\alpha \in (0, 1)$ there is a constant $c=c(\alpha) > 0$ such that any planar $(\alpha, 2\alpha)$-Furstenberg set has Hausdorff dimension at least $2\alpha + c$. This improves several previous bounds, in particular…

We derive estimates of the Babu\u{s}ka-Brezzi inf-sup constant $\beta$ for two-dimensional incompressible flow in a periodic channel with one flat boundary and the other given by a periodic, Lipschitz continuous function $h$. If $h$ is a…

Let $f$ be an expansive Lorenz map on $[0,1]$ and $c$ be the critical point. The survivor set we are discussing here is denoted as $S^+_{f}(a,b):=\{x\in[0,1]:f(b)\leq f^{n}(x) \leq f(a)\ \forall n\geq0\}$, where the hole $(a,b)\subseteq…

Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than…

Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding…