On Fourier dimension and Salem sets

Abstract

Fourier dimension is connected to the decay of the Fourier transform of measures through energy integrals and is bounded by the Hausdorff dimension. Study of the applications of expressing energy integrals in terms of the Fourier transform and the Fourier series dates to the 1960s works of Kahane and Salem, and Carleson. The sets with equal Fourier and Hausdorff dimensions are called Salem sets, named after the Greek mathematician Raphaël Salem who first gave a construction of such sets in 1951. Fourier transforms of measures have applications in, for example, number theory, complex analysis, and operator theory.

There are two main goals in this thesis. In Chapter 3, we introduce the Fourier dimension and prove some of its properties. Results regarding the additivity and stability of the Fourier dimension are considered, with comparison to the Hausdorff dimension. The second goal, and the bigger part of this thesis, is to introduce Salem sets which we do in Chapter 4. These include some deterministic sets, however, emphases will be put on probabilistic examples with a focus on the images of sets and measures under some random mappings.

In Chapter 2, we go through the preliminaries including the notation, definitions, and the fundamental results used throughout this work. They concern measure theory, Fourier analysis, and probability theory, and can be found in most of the textbooks on the topics. More specific results are given as a part of the proof when required.