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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 $A\subseteq {ℝ}^{3}$ is a Borel set of Hausdorff dimension $\mathrm{dim}A\in \left(\frac{3}{2},\frac{5}{2}\right)$, then for a.e. $𝜃\in \left[0,2\pi \right)$ the projection ${\pi }_{𝜃}\left(A\right)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}\left(\mathrm{cos}𝜃,\mathrm{sin}𝜃,1\right)$ satisfies $\mathrm{dim}{\pi }_{𝜃}\left(A\right)\ge \mathrm{max}\left\{\frac{4}{9}\mathrm{dim}A+\frac{5}{6},\frac{1}{3}\left(2\mathrm{dim}A+1\right)\right\}$. 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 $\mathrm{dim}A\in \left(\frac{3}{2},\frac{5}{2}\right)$. 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.

##### Keywords
orthogonal projections, Hausdorff dimension, Fourier transform
##### Mathematical Subject Classification 2010
Primary: 42B10, 28E99