On the Fourier dimension of (d, k)-sets and Kakeya sets with restricted directions

Autor: Jonathan M. Fraser, Terence L. J. Harris, Nicholas G. Kroon
Přispěvatelé: EPSRC, The Leverhulme Trust, University of St Andrews. Pure Mathematics, University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra
Rok vydání: 2022
Předmět:
Zdroj: Mathematische Zeitschrift. 301:2497-2508
ISSN: 1432-1823
0025-5874
DOI: 10.1007/s00209-022-02971-3
Popis: A $(d,k)$-set is a subset of $\mathbb{R}^d$ 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 $\Gamma$. In this setting our estimates depend on the Hausdorff dimension of $\Gamma$ and can sometimes be improved if additional geometric properties of $\Gamma$ are assumed. We are led to consider cones and prove that the cone in $\mathbb{R}^{d+1}$ has Fourier dimension $d-1$, which may be of interest in its own right.
Comment: 13 pages. This revised version subsumes arXiv:2108.05771, contains a new result about the Fourier dimension of cones and includes a new co-author
Databáze: OpenAIRE
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