Hausdorff dimension, projections, and the Fourier transform
type: Item
instance of: scholarly article
title: Hausdorff dimension, projections, and the Fourier transform
zbMATH Open document ID: 1049.28007
DOI: 10.5565/PUBLMAT_48104_01
author: Pertti Mattila
published in: Publicacions Matemàtiques
publication date: 2004-06-14T00:00:00Z
full work available at URL: https://eudml.org/doc/41489
review text: 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.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 2074328
zbMATH Keywords: Cantor sets
zbMATH Keywords: Besicovitch sets
zbMATH Keywords: potential theory
zbMATH Keywords: decay
zbMATH Keywords: Fourier transform
zbMATH Keywords: Kakeya problem
reviewed by: Terence Tao
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2074328