The Fourier Analysis and Hausdorff Dimension

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.Preface; Acknowledgements; 1. Introduction; 2. Measure theoretic preliminaries; 3. Fourier transforms; 4. Hausdorff dimension of projections and distance sets; 5. Exceptional projections and Sobolev dimension; 6. Slices of measures and intersections with planes; 7. Intersections of general sets and measures; 8. Cantor measures; 9. Bernoulli convolutions; 10. Projections of the four-corner Cantor set; 11. Besicovitch sets; 12. Brownian motion; 13. Riesz products; 14. Oscillatory integrals (stationary phase) and surface measures; 15. Spherical averages and distance sets; 16. Proof of the Wolff-Erdogan Theorem; 17. Sobolev spaces, Schrodinger equation and spherical averages; 18. Generalized projections of Peres and Schlag; 19. Restriction problems; 20. Stationary phase and restriction; 21. Fourier multipliers; 22. Kakeya problems; 23. Dimension of Besicovitch sets and Kakeya maximal inequalities; 24. (n, k) Besicovitch sets; 25. Bilinear restriction; References; List of basic notation; Author index; Subject index.