Quasisymmetric Maps on Kakeya Sets

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Orponen , T 2017 , ' Quasisymmetric Maps on Kakeya Sets ' , International Mathematics Research Notices , no. 11 , pp. 3413-3425 . https://doi.org/10.1093/imrn/rnw131

Title: Quasisymmetric Maps on Kakeya Sets
Author: Orponen, Tuomas
Contributor organization: Department of Mathematics and Statistics
Date: 2017-06
Language: eng
Number of pages: 13
Belongs to series: International Mathematics Research Notices
ISSN: 1073-7928
DOI: https://doi.org/10.1093/imrn/rnw131
URI: http://hdl.handle.net/10138/307633
Abstract: I show that L-p - L-q estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known L-p - L-q estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in R-n is at least max{(n+2)/2, (4n+3)/7}. Moreover, if f is a quasisymmetry from a Kakeya set K subset of R-n onto any at most n-dimensional metric space, the f - image of a.e. line segment inside K has dimension at most min{2n/(n + 2), 7n/(4n+3)}. The Kakeya maximal function conjecture implies that the bounds can be improved to n and 1, respectively.
Subject: 111 Mathematics
Peer reviewed: Yes
Rights: other
Usage restriction: openAccess
Self-archived version: acceptedVersion

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