A Survey on the Fourier Dimension

This article is about various aspects of the Fourier dimension and its variants. One aspect is to relate, and contrast the Fourier dimension with the Hausdorff dimension. Moreover we will present some questions where the Fourier dimension can be successfully applied. This includes uniform distribution problems and questions from geometric measure theory like the occurrence of Salem sets. There have been several similar but different notions of the Fourier dimension subject to different applications. We will argue that these various notions are indeed different and also do not behave like a regular dimension-like quantity. We also will give an alternative more regular definition that still reflects most of the important properties that are needed for applications.