On the Fourier dimension and a modification
(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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/278a59ff-fcea-47ad-a683-e1e881bf1230
- author
- Ekström, Fredrik ^{LU} ; Persson, Tomas ^{LU} and Schmeling, Jörg ^{LU}
- organization
- publishing date
- 2015
- 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}, }