The main object of this book is the interplay between geometric measure theory and Fourier analysis on ℝn. The emphasis will be more on the first in the sense that on several occasions we look for the best known results in geometric measure theory while our goals in Fourier analysis will usually be much more modest. We shall concentrate on those parts of Fourier analysis where Hausdorff dimension plays a role. Much more between geometric measure theory and Fourier analysis has been and is going on. Relations between singular integrals and rectifiability have been intensively studied formore than two decades; see the books David and Semmes [1993], Mattila [1995] and Tolsa [2014], the survey Volberg and Eiderman [2013], and Nazarov, Tolsa and Volberg [2014] for recent break-through results. Relations between harmonic measure, partial differential equations (involving a considerable amount of Fourier analysis) and rectifiability have recently been very actively investigated by many researchers; see, for example, Kenig and Toro [2003], Hofmann, Mitrea and Taylor [2010], Hofmann, Martell and Uriarte-Tuero [2014], and the references given therein.

In this book there are two main themes. Firstly, the Fourier transform is a powerful tool on geometric problems concerning Hausdorff dimension, and we shall givemany applications. Secondly, some basic problems ofmodern Fourier analysis, in particular those concerning restriction, are related to geometric measure theoretic Kakeya (or Besicovitch) type problems. We shall discuss these in the last part of the book. We shall also consider various particular constructions of measures and the behaviour of their Fourier transforms.

The contents of this book can be divided into four parts.

1. PART I Preliminaries and some simpler applications of the Fourier transform.

2. PART II Specific constructions.

3. PART III Deeper applications of the Fourier transform.

4. PART IV Fourier restriction and Kakeya type problems.

Parts I and III are closely linked together. They are separated by Part II only because much of the material in Part III is rather demanding and Part II might be more easily digestible. In any case, the reader may jump over Part II without any problems. On the other hand, the sections of Part II are essentially independent of each other and only rely on Chapters 2 and 3. Part IV is nearly independent of the others.