Related papers: Attainable forms of Assouad spectra

Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $\phi$ be a…

A graphon that is defined on $[0,1]^d$ and is H\"older$(\alpha)$ continuous for some $d\ge2$ and $\alpha\in(0,1]$ can be represented by a graphon on $[0,1]$ that is H\"older$(\alpha/d)$ continuous. We give examples that show that this…

Let $f(x,y) \not\equiv 0$ be a real-analytic planar function. We show that, for almost every $R>0$ there exists an analytic 1-parameter family of vector fields $X_{\lambda}$ which has $\{f(x,y)=0\} \cap \bar{B_R((0,0))}$ as a limit periodic…

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated…

We establish $\frac{1}{2}$-H\"older continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schr\"odinger operators under suitable boundary conditions and exponentially decaying small potentials. These…

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost-additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated…

We present results on the broadband nature of the power spectrum $S(\omega)$, $\omega\in(0,2\pi)$, for a large class of nonuniformly expanding maps with summable and nonsummable decay of correlations. In particular, we consider a class of…

For a complex variable $s$ and real parameters $a$ and $\lambda$ with $a>0$, let $\phi(s,a,\lambda)$ denote the Lerch zeta-function with a complex variable, $\phi^{\ast}(s,a,\lambda)$ a slight modification of $\phi(s,a,\lambda)$ defined by…

The spectrum of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which…

For a function $F$ represented as $F(x)=\sum_{n=0}^\infty{f_n (x) e^{2 \pi i \lambda_n x}},$ where each $f_n$ satisfies $\operatorname{spec}(f_n) \subset [0, 1]$ and $(\lambda_n)_{n\geq 0}\subset \mathbb{R}_+$ is a lacunary sequence, we…

Let ($M$, $\Omega$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^l$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose…

Let ${\mathcal F}_\lambda(\mathbb{S}^n)$ be the space of tensor densities on $\mathbb{S}^n$ of degree $\lambda$. We consider this space as an induced module of the nonunitary spherical series of the group $\mathrm{SO}_0(n+1,1)$ and classify…

Suppose $q$ is a fixed odd prime power, $F(\vec{x})$ is a non-degenerate quadratic form over $\mathbb{F}_q[t]$ of discriminant $\Delta$ in $d\geq 5$ variables $\vec{x}$, and $f,g\in\mathbb{F}_q[t]$,…

Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each…

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice $\Lambda \subseteq \mathbb{R}^d$ inside the $d$-sphere of radius $R$. In contrast to…

We study discrete Schroedinger operators $(H_{\alpha,\theta}\psi)(n)= \psi(n-1)+\psi(n+1)+f(\alpha n+\theta)\psi(n)$ on $l^2(Z)$, where $f(x)$ is a real analytic periodic function of period 1. We prove a general theorem relating the measure…

Let $n\in\mathbb{N}$ and let $\Theta \subset \{1,\dots,n\}$ be a non-empty subset. We prove that if $\Theta$ contains an odd integer, then any $P_\Theta$-Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ is virtually isomorphic to a free group…

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that $\Omega$ is spectral if and only if it can tile the space by…

Let $Z_\alpha$ be a positive $\alpha-$stable random variable and $r\in{\bf R}.$ We show the existence of an unbounded open domain $D$ in $[1/2,1]\times{\bf R}$ with a cusp at $(1/2,-1/2)$, characterized by the complete monotonicity of the…

Mutually orthogonal frequency squares (MOFS) of type $F(m\lambda;\lambda)$ generalize the structure of mutually orthogonal Latin squares: rather than each of $m$ symbols appearing exactly once in each row and in each column of each square,…