Related papers: Self-similar measures with unusual local dimension…

In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions. We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that,…

The paper introduces a general method to construct conformal measures for a local homeomorphism on a locally compact non-compact Hausdorff space, subject to mild irreducibility-like conditions. Among others the method is used to give…

In this paper we show that if $\mu$ is any locally and uniformly $\alpha$-dimensional measure supported on a $\alpha$-quasi-regular set $E$, then $L^2(\mu)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor…

A metric measure space $(X,d,\mu)$ is said to satisfy the strong annular decay condition if there is a constant $C>0$ such that $$ \mu\big(B(x,R)\setminus B(x,r)\big)\leq C\, \frac{R-r}{R}\, \mu (B(x,R)) $$ for each $x\in X$ and all $0<r…

As is well known, quantum mechanical behavior cannot, in general, be simulated by a local hidden variables model. Most -if not all- the proofs of this incompatibility refer to the correlations which arise when each of two (or more) systems…

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $\Phi=\left\{ \varphi_{i}\right\} _{i\in\Lambda}$ be an iterated function system on $I:=[0,1]$…

We prove that a self similar measure is absolutely continuous providing that it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations of the self similar…

Let $\{S_i\}_{i\in \Lambda}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(\Sigma,\sigma)$ denote the two-sided full shift over the alphabet $\Lambda$, and $\pi:\Sigma\to {\Bbb R}^d$ be the coding map…

We consider families of geometries of D--dimensional space, described by a finite number of parameters. Starting from the De Witt metric we extract a unique integration measure which turns out to be a geometric invariant, i.e. independent…

We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish.…

This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is a set function which is finitely additive on the collection of open and compact sets, inner regular on open sets, and…

Let $\mu$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $\mu$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support.…

An inhomogeneous fractal set is one which exhibits different scaling behaviour at different points. The Assouad dimension of a set is a quantity which finds the `most difficult location and scale' at which to cover the set and its…

Measurable sets are defined as those locally approximable, in a certain sense, by sets in the given algebra (or ring). A corresponding measure extension theorem is proved. It is also shown that a set is locally approximable in the mentioned…

Consider an iterated function system consisting of similarities on the complex plane of the form $g_{i}(z) = \lambda_i z + t_i,\ \lambda_i, t_i \in \mathbb{C},\ |\lambda_i|<1, i=1,\ldots, k$. We prove that for almost every choice of…

This article proves the existence and regularity of weak solutions for a class of mixed local-nonlocal problems with singular nonlinearities. We examine both the purely singular problem and perturbed singular problems. A central…

In this paper we develop a general framework of badly approximable points in a metric space $X$ equipped with a $\sigma$-finite doubling Borel regular measure $\mu$. We establish that under mild assumptions the $\mu$-measure of the set of…

R. Kaufman and M. Tsujii proved that the Fourier transform of self-similar measures has a power decay outside of a sparse set of frequencies. We present a version of this result for homogeneous self-similar measures, with quantitative…

There are two definitions of the measurable functional on the topological vector space: as a linear and measurable real-valued function and as a pointwise limit of the sequence of the continious linear functionals. In general case they are…

We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-spectrum are other special examples of these intermediate dimensions. These dimensions are…