Improved bounds for restricted projection families via weighted Fourier restriction

Harris, Terence L. J.

doi:10.2140/apde.2022.15.1655

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Improved bounds for restricted projection families via weighted Fourier restriction

Terence L. J. Harris

Vol. 15 (2022), No. 7, 1655–1701

DOI: 10.2140/apde.2022.15.1655

Abstract

It is shown that if $A \subseteq ℝ^{3}$ is a Borel set of Hausdorff dimension $\dim A \in (\frac{3}{2}, \frac{5}{2})$ , then for a.e. $𝜃 \in [0, 2 π)$ the projection $π_{𝜃} (A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}} (\cos 𝜃, \sin 𝜃, 1)$ satisfies $\dim π_{𝜃} (A) \geq \max {\frac{4}{9} \dim A + \frac{5}{6}, \frac{1}{3} (2 \dim A + 1)}$ . This improves the bounds of Oberlin and Oberlin (J. Geom. Anal. 25:3 (2015), 1476–1491) and of Orponen and Venieri (Int. Math. Res. Not. 2020:19 (2020), 5797–5813) for $\dim A \in (\frac{3}{2}, \frac{5}{2})$ . More generally, a weaker lower bound is given for families of planes in $ℝ^{3}$ parametrised by curves in $S^{2}$ with nonvanishing geodesic curvature.

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Keywords

orthogonal projections, Hausdorff dimension, Fourier transform

Mathematical Subject Classification 2010

Primary: 42B10, 28E99

Milestones

Received: 9 March 2020

Revised: 5 February 2021

Accepted: 19 March 2021

Published: 5 December 2022

Authors

Terence L. J. Harris
Department of Mathematics Cornell University Ithaca, NY United States	Department of Mathematics University of Illinois Urbana, IL United States

Vol. 15, No. 7, 2022