The Fourier spectrum and sumset type problems

Jonathan Fraser*

*Corresponding author for this work

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Abstract

We introduce and study the Fourier spectrum which is a continuously parametrised family of dimensions living between the Fourier dimension and the Hausdorff dimension for both sets and measures. We establish some fundamental theory and motivate the concept via several applications, especially to sumset type problems. For example, we study dimensions of convolutions and sumsets, and solve the distance set problem for sets satisfying certain Fourier analytic conditions.
Original languageEnglish
JournalMathematische Annalen
DOIs
Publication statusPublished - 5 Apr 2024

Keywords

  • Fourier spectrum
  • Fourier transform
  • Fourier dimension
  • Sobolev dimension
  • Hausdorff dimension
  • Convolution
  • Distance set
  • Sunset

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