On the Fourier dimension and a modification
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 a measure has modified Fourier dimension greater than or equal to if and only if it annihilates all sets with modified Fourier dimension less than .
Cite
@article{arxiv.1406.1480,
title = {On the Fourier dimension and a modification},
author = {Fredrik Ekström and Tomas Persson and Jörg Schmeling},
journal= {arXiv preprint arXiv:1406.1480},
year = {2016}
}
Comments
v2: Added some remarks in the introduction and after Example 6. v3: Revised the introduction, strengthened Lemma 6, added Proposition 5 and Example 8. To appear in Journal of Fractal Geometry