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Transversal families
of nonlinear projections and generalizations of Favard
length
Rosemarie Bongers and Krystal Taylor |
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Vol. 16 (2023), No. 1, 279–308
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Abstract |
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Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a transversality condition. We establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four-corner Cantor set, first established by Cladek, Davey, and Taylor. |
Keywords
nonlinear projections, transversality, Favard length,
fractals, transversal projections, Buffon needle, energy,
rate of decay, radial projections, curve projections,
rectifiability
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Mathematical Subject Classification
Primary: 28A75, 28A80, 57N75
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Milestones
Received: 17 May 2021
Revised: 4 February 2022
Accepted: 22 March 2022
Published: 14 April 2023
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Authors |
| Rosemarie Bongers | |
| Department of Mathematics Harvard University Cambridge, MA United States |
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| Krystal Taylor | |
| Department of Mathematics The Ohio State University Columbus, OH United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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