Related papers: On dimension and regularity of bundle measures

We are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a…

Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of…

Let $P\to M$ be a principal bundle. Consider a sequence of metrics on $P$ obtained by re-scaling the fibers to points. The Gromov-Hausdorff limit of the tangent bundles over these principal bundles with their Sasaki metric is seen herein to…

These are notes of a talk based on the work arXiv:1212.3630 joint with A. Aizenbud. Let V be a finite-dimensional vector space over a local field F of characteristic 0. Let f be a function on V of the form $f(x)= \psi (P(x))$, where P is a…

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's…

We show positivity of the lower Fourier dimension for equilibrium states of nonlinear, area preserving, Axiom A diffeomorphisms on surfaces. To do so, we use the sum-product phenomenon to reduce Fourier decay to the study of some temporal…

Consider an operator that takes the Fourier transform of a discrete measure supported in $\mathcal{X}\subset[-\frac 12,\frac 12)^d$ and restricts it to a compact $\Omega\subset\mathbb{R}^d$. We provide lower bounds for its smallest singular…

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher…

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 $X$ be a compact K\"ahler manifold and let $(L, \varphi)$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension of pseudo-effective line bundles with singular metrics, and then discuss the properties…

We show that the Fourier-Laplace transform of an irreducible regular differential system on the Riemann sphere underlies, when one only considers the part at finite distance, a polarizable regular twistor $\mathcal{D}$-module. The…

We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed…

A well-known theorem of J.E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal…

Given a compact set of real numbers, a random $C^{m + \alpha}$-diffeomorphism is constructed such that the image of any measure concentrated on the set and satisfying a certain condition involving a real number $s$, almost surely has…

We translate Akin's notion of {\it good} (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups),…

We describe a setup for obtaining uncertainty relations for arbitrary pairs of observables related by Fourier transform. The physical examples discussed here are standard position and momentum, number and angle, finite qudit systems, and…

We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower bounds on the Hausdorff dimension of $\mu$. As an application, we give an elementary and…

A $(d,k)$-set is a subset of $\mathbb{R}^d$ containing a $k$-dimensional unit ball of all possible orientations. Using an approach of D.~Oberlin we prove various Fourier dimension estimates for compact $(d,k)$-sets. Our main interest is in…

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as…

Given a Kaehlerian holomorphic fiber bundle whose fiber is a compact homogeneous Kaehler manifold, we describe the perturbed Hermitian-Einstein equations relative to certain holomorphic vector bundles. With respect to special metrics on the…