Projects per year
Abstract
A (d, k)-set is a subset of ℝd containing a k-dimensional unit ball of all possible orientations. Using an approach of D. Oberlin we prove various Fourier dimension estimates for compact (d, k)-sets. Our main interest is in restricted (d, k)-sets, where the set only contains unit balls with a restricted set of possible orientations Γ. In this setting our estimates depend on the Hausdorff dimension of Γ and can sometimes be improved if additional geometric properties of Γ are assumed. We are led to consider cones and prove that the cone in ℝd+1 has Fourier dimension d−1, which may be of interest in its own right.
Original language | English |
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Number of pages | 12 |
Journal | Mathematische Zeitschrift |
Volume | First Online |
Early online date | 24 Feb 2022 |
DOIs | |
Publication status | E-pub ahead of print - 24 Feb 2022 |
Keywords
- Fourier dimension
- Kakeya set
- (d, k)-set
- Hausdorff dimension
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Dive into the research topics of 'On the Fourier dimension of (d,k)-sets and Kakeya sets with restricted directions'. Together they form a unique fingerprint.Projects
- 2 Finished
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New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
1/09/19 → 31/01/23
Project: Standard
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Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
1/02/18 → 11/06/21
Project: Standard