Related papers: Dimension spectrum for a nonconventional ergodic a…
We determine the Hausdorff dimension of sets of irrationals in $(0,1)$ whose partial quotients in semi-regular continued fractions obey certain restrictions and growth conditions. This result substantially generalizes that of the second…
For any beta-shift $(X_\beta,\sigma)$ on two symbols, i.e., the symbolic coding of the beta-map for $1<\beta\leq2$, we give an exact formula for the Hausdorff dimension $\dim_{H} \Lambda_{\alpha(t)}$ as a function of $t\in\mathbb{R}$, where…
In this paper, a new statistic feature of the discrete short-time amplitude spectrum is discovered by experiments for the signals of unvoiced pronunciation. For the random-varying short-time spectrum, this feature reveals the relationship…
We are going to introduce a new algebraic, analytic structure that is a kind of generalization of the Hausdorff dimension and measure. We give many examples and study the basic properties and relations of such systems.
To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy…
We study the Hausdorff dimension of Poissonian cutout sets defined via inhomogeneous intensity measures on Ahlfors-regular metric spaces. We obtain formulas for the Hausdorff dimension of such cutouts in self-similar and self-conformal…
In 1996 Y. Kifer obtained a variational formula for the Hausdorff dimension of the set of points for which the frequencies of the digits in the Cantor series expansion is given. In this note we present a slightly different approach to this…
We investigate the set of $x \in S^1$ such that for every positive integer $N$, the first $N$ points in the orbit of $x$ under rotation by irrational $\theta$ contain at least as many values in the interval $[0,1/2]$ as in the complement.…
We show the relevance of a multifractal-type analysis for pointwise convergence and divergence properties of wavelet series: Depending on the sequence space which the wavelet coefficients sequence belongs to, we obtain deterministic upper…
Two spectral triples are introduced for a class of fractals in R^n. The definitions of noncommutative Hausdorff dimension and noncommutative tangential dimensions, as well as the corresponding Hausdorff and Hausdorff-Besicovitch functionals…
Let $\alpha$ be an irrational real number. We show that the set of $\epsilon$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (\alpha) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| q\alpha -x \| \geq \varepsilon \} \]…
For piecewise monotone interval maps we look at Birkhoff spectra for regular potential functions. This means considering the Hausdorff dimension of the set of points for which the Birkhoff average of the potential takes a fixed value. In…
A useful approach for analysing multiple time series is via characterising their spectral density matrix as the frequency domain analog of the covariance matrix. When the dimension of the time series is large compared to their length,…
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…
The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete…
For an irrational number $x\in [0,1)$, let $x=[a\_1(x), a\_2(x),\cdots]$ be its continued fraction expansion. Let $\psi : \mathbb{N} \rightarrow \mathbb{N}$ be a function with $\psi(n)/n\to \infty$ as $n\to\infty$. The (upper, lower) fast…
For Borel subsets A and B of the Euclidean n-space the intersection of A with generic rotations and translations of B has often Hausdorff dimension at least dim A + dim B - n. Estimates for the exceptional set of rotations are derived.
We compute the massless spectra of a set of flux vacua of the heterotic string. The vacua we study include well-known non-Kahler T^2-fibrations over K3 with SU(3) structure and intrinsic torsion. Following gauged linear sigma models of…
The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the…
This article surveys the $\theta$-intermediate dimensions that were introduced recently which provide a parameterised continuum of dimensions that run from Hausdorff dimension when $\theta=0$ to box-counting dimensions when $\theta=1$. We…