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Fourier dimension of random images

Ekström, Fredrik LU (2016) In Arkiv for Matematik 54(2). p.455-471
Abstract

Given a compact set of real numbers, a random Cm + α-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 Fourier dimension greater than or equal to s/ (m+ α). This is used to show that every Borel subset of the real numbers of Hausdorff dimension s is Cm + α-equivalent to a set of Fourier dimension greater than or equal to s/ (m+ α). In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under Cm-diffeomorphisms for any m.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Arkiv for Matematik
volume
54
issue
2
pages
17 pages
publisher
Springer
external identifiers
  • Scopus:84979587640
ISSN
0004-2080
DOI
10.1007/s11512-016-0237-3
language
English
LU publication?
yes
id
08c91cf1-fc98-460c-a304-893c220edf5a
date added to LUP
2016-10-17 07:37:56
date last changed
2016-10-17 07:37:56
@misc{08c91cf1-fc98-460c-a304-893c220edf5a,
  abstract     = {<p>Given a compact set of real numbers, a random C<sup>m</sup> <sup>+</sup>         <sup>α</sup>-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 Fourier dimension greater than or equal to s/ (m+ α). This is used to show that every Borel subset of the real numbers of Hausdorff dimension s is C<sup>m</sup> <sup>+</sup> <sup>α</sup>-equivalent to a set of Fourier dimension greater than or equal to s/ (m+ α). In particular every Borel set is diffeomorphic to a Salem set, and the Fourier dimension is not invariant under C<sup>m</sup>-diffeomorphisms for any m.</p>},
  author       = {Ekström, Fredrik},
  issn         = {0004-2080},
  language     = {eng},
  month        = {10},
  number       = {2},
  pages        = {455--471},
  publisher    = {ARRAY(0x81d7e88)},
  series       = {Arkiv for Matematik},
  title        = {Fourier dimension of random images},
  url          = {http://dx.doi.org/10.1007/s11512-016-0237-3},
  volume       = {54},
  year         = {2016},
}