54,391 research outputs found
Describability via ubiquity and eutaxy in Diophantine approximation
We present a comprehensive framework for the study of the size and large
intersection properties of sets of limsup type that arise naturally in
Diophantine approximation and multifractal analysis. This setting encompasses
the classical ubiquity techniques, as well as the mass and the large
intersection transference principles, thereby leading to a thorough description
of the properties in terms of Hausdorff measures and large intersection classes
associated with general gauge functions. The sets issued from eutaxic sequences
of points and optimal regular systems may naturally be described within this
framework. The discussed applications include the classical homogeneous and
inhomogeneous approximation, the approximation by algebraic numbers, the
approximation by fractional parts, the study of uniform and Poisson random
coverings, and the multifractal analysis of L{\'e}vy processes.Comment: 94 pages. Notes based on lectures given during the 2012 Program on
Stochastics, Dimension and Dynamics at Morningside Center of Mathematics, the
2013 Arithmetic Geometry Year at Poncelet Laboratory, and the 2014 Spring
School in Analysis held at Universite Blaise Pasca
Hausdorff and packing dimensions of the images of random fields
Let be a random field with values in
. For any finite Borel measure and analytic set
, the Hausdorff and packing dimensions of the image
measure and image set are determined under certain mild
conditions. These results are applicable to Gaussian random fields,
self-similar stable random fields with stationary increments, real harmonizable
fractional L\'{e}vy fields and the Rosenblatt process.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ244 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On Fourier analytic properties of graphs
We study the Fourier dimensions of graphs of real-valued functions defined on
the unit interval [0,1]. Our results imply that the graph of the fractional
Brownian motion is almost surely not a Salem set, answering in part a question
of Kahane from 1993, and that the graph of a Baire typical function in C[0,1]
has Fourier dimension zero.Comment: 11 pages, 1 figure; references added and typos corrected in v2; to
appear in Int. Math. Res. Not. IMR
Packing-Dimension Profiles and Fractional Brownian Motion
In order to compute the packing dimension of orthogonal projections
Falconer and Howroyd (1997) introduced a family of packing dimension profiles
that are parametrized by real numbers . Subsequently,
Howroyd (2001) introduced alternate -dimensional packing dimension profiles
\hbox{{\rm P}\dim}_s and proved, among many other things, that
\hbox{{\rm P}\dim}_s E={\rm Dim}_s E for all integers and all
analytic sets . The goal of this article is to prove that
\hbox{{\rm P}\dim}_s E={\rm Dim}_s E for all real numbers and
analytic sets . This answers a question of Howroyd (2001, p.
159). Our proof hinges on a new property of fractional Brownian motion
Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates
With a view toward fractal spaces, by using a Korevaar-Schoen space approach,
we introduce the class of bounded variation (BV) functions in a general
framework of strongly local Dirichlet spaces with a heat kernel satisfying
sub-Gaussian estimates. Under a weak Bakry-\'Emery curvature type condition,
which is new in this setting, this BV class is identified with a heat semigroup
based Besov class. As a consequence of this identification, properties of BV
functions and associated BV measures are studied in detail. In particular, we
prove co-area formulas, global Sobolev embeddings and isoperimetric
inequalities. It is shown that for nested fractals or their direct products the
BV class we define is dense in . The examples of the unbounded Vicsek set,
unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.Comment: The notes arXiv:1806.03428 will be divided in a series of papers.
This is the third paper. v2: Final versio
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