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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
Abstract

It is shown that if A 3 is a Borel set of Hausdorff dimension dim A (3 2, 5 2), then for a.e. 𝜃 [0,2π) the projection π𝜃(A) of A onto the 2-dimensional plane orthogonal to 1 2(cos 𝜃,sin 𝜃,1) satisfies dim π𝜃(A) max {4 9 dim A + 5 6, 1 3(2dim 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 (3 2, 5 2). More generally, a weaker lower bound is given for families of planes in 3 parametrised by curves in S2 with nonvanishing geodesic curvature.

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