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On the Fourier dimension and a modification

Ekström, Fredrik LU ; Persson, Tomas LU and Schmeling, Jörg LU (2015) In Journal of Fractal Geometry 2(3). p.309-337
Abstract
We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that for each s there is a... (More)
We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that for each s there is a class of sets such that a measure has modied Fourier dimension greater than or equal to s if and only if it annihilates all sets in the class. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Stability of Fourier dimension, modified Fourier dimension
in
Journal of Fractal Geometry
volume
2
issue
3
pages
309 - 337
DOI
10.4171/JFG/23
language
English
LU publication?
yes
id
278a59ff-fcea-47ad-a683-e1e881bf1230
date added to LUP
2017-03-06 16:23:01
date last changed
2017-05-02 11:07:46
@article{278a59ff-fcea-47ad-a683-e1e881bf1230,
  abstract     = {We give a sufficient condition for the Fourier dimension of a countable union of sets to equal the supremum of the Fourier dimensions of the sets in the union, and show by example that the Fourier dimension is not countably stable in general. A natural approach to finite stability of the Fourier dimension for sets would be to try to prove that the Fourier dimension for measures is finitely stable, but we give an example showing that it is not in general. We also describe some situations where the Fourier dimension for measures is stable or is stable for all but one value of some parameter. Finally we propose a way of modifying the definition of the Fourier dimension so that it becomes countably stable, and show that for each s there is a class of sets such that a measure has modied Fourier dimension greater than or equal to s if and only if it annihilates all sets in the class.},
  author       = {Ekström, Fredrik and Persson, Tomas and Schmeling, Jörg},
  keyword      = {Stability of Fourier dimension,modified Fourier dimension},
  language     = {eng},
  number       = {3},
  pages        = {309--337},
  series       = {Journal of Fractal Geometry},
  title        = {On the Fourier dimension and a modification},
  url          = {http://dx.doi.org/10.4171/JFG/23},
  volume       = {2},
  year         = {2015},
}