During the past two decades there has been active interplay between
geometric measure theory and Fourier analysis. This book describes
part of that development, concentrating on the relationship between
the Fourier transform and Hausdorff dimension. The main topics
concern applications of the Fourier transform to geometric problems
involving Hausdorff dimension, such as Marstrand type projection
theorems and Falconer's distance set problem, and the role of
Hausdorff dimension in modern Fourier analysis, especially in
Kakeya methods and Fourier restriction phenomena. The discussion
includes both classical results and recent developments in the
area. The author emphasises partial results of important open
problems, for example, Falconer's distance set conjecture, the
Kakeya conjecture and the Fourier restriction conjecture.
Essentially self-contained, this book is suitable for graduate
students and researchers in mathematics.
General
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