The Fourier spectrum and sumset type problems

Jonathan Fraser

doi:10.1007/s00208-024-02843-7

The Fourier spectrum and sumset type problems

Jonathan Fraser^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 language	English
Journal	Mathematische Annalen
DOIs	https://doi.org/10.1007/s00208-024-02843-7
Publication status	Published - 5 Apr 2024

Keywords

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

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10.1007/s00208-024-02843-7Licence: CC BY

Fraser_2024_MA_Fourier-spectrum_CC
Copyright © The Author(s) 2024. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Final published version, 542 KBLicence: CC BY

Sabbatical Research Grant 2021: Sabbatical Research Grant
Fraser, J. (PI)
1/08/21 → 31/07/22
Project: Standard
New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
The Leverhulme Trust
1/09/19 → 31/01/23
Project: Standard
Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard

Cite this

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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.",

keywords = "Fourier spectrum, Fourier transform, Fourier dimension, Sobolev dimension, Hausdorff dimension, Convolution, Distance set, Sunset",

author = "Jonathan Fraser",

note = "Funding: EPSRC Standard Grant (EP/R015104/1), a Leverhulme Trust Research Project Grant (RPG-2019-034) and an RSE Sabbatical Research Grant (70249). ",

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AB - 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.

KW - Fourier spectrum

KW - Fourier transform

KW - Fourier dimension

KW - Sobolev dimension

KW - Hausdorff dimension

KW - Convolution

KW - Distance set

KW - Sunset

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The Fourier spectrum and sumset type problems

Abstract

Keywords

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Projects

Sabbatical Research Grant 2021: Sabbatical Research Grant

New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Cite this