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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.
The focus of this book is geometric properties of general sets and measures in Euclidean spaces. Applications of this theory include fractal-type objects, such as strange attractors for dynamical systems, and those fractals used as models in the sciences. The author provides a firm and unified foundation for the subject and develops all the main tools used in its study, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Besicovitch-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of Euclidean space possessing many of the properties of smooth surfaces.
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