Hausdorff dimension, projections, and the Fourier transform
This is a thorough and beautiful survey of the subject of Hausdorff dimension and the behavior of sets under generic projection mappings. A typical problem discussed in this survey is that of determining the behavior of the projection of the Cantor dust set \(C_{1/4} \times C_{1/4} \subset R^2\) with respect to some angle \(\theta\); while it is known that such projections almost surely have Lebesgue measure zero, and some very weak quantitative bounds (whose proof is sketched in this survey) are known on the \(\delta\)-neighborhoods of such projections, there are still many unsolved questions about this very explicit problem. The author also surveys the closely related issue of the decay properties of the Fourier transform of measures of finite energy, and the connection to the Falconer distance conjecture; he also addresses the Kakeya problem and some known results, as well as the theory of generalized projections as studied by Peres and Schlag, and related questions of visibility of sets around a generic element of the set. The author concludes with a nice list of open problems in the subject.
scientific article; zbMATH DE number 2074328