Abstract
We establish Marstrand-type projection theorems for orthogonal projections along geodesics onto m-dimensional subspaces of the hyperbolic n-space by a geometric argument. Moreover, we obtain a Besicovitch–Federer type characterization of purely unrectifiable sets in terms of these hyperbolic orthogonal projections.
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Balogh, Z.M., Durand-Cartagena, E., Fässler, K., Mattila, P., Tyson, J.T.: The effect of projections on dimension in the Heisenberg group. Rev. Mat. Iberoam. 29(2), 381–432 (2013)
Balogh, Z.M., Fässler, K., Mattila, P., Tyson, J.T.: Projection and slicing theorems in Heisenberg groups. Adv. Math. 231(2), 569–604 (2012)
Balogh, Z.M., Iseli, A.: Marstrand type projection theorems for normed spaces. To appear in J. Fractal Geom. (2018). arXiv:1802.10563
Balogh, Z.M., Iseli, A.: Dimensions of projections of sets on Riemannian surfaces of constant curvature. Proc. Am. Math. Soc. 144(7), 2939–2951 (2016)
Benedetti, R., Petronio, C.: Lectures on Hyperbolic Geometry. Universitext. Springer, Berlin (1992)
Besicovitch, A.S.: On the fundamental geometrical properties of linearly measurable plane sets of points (III). Math. Ann. 116(1), 349–357 (1939)
Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)
Chen, C.: Restricted families of projections and random subspaces. (2017). arXiv:1706.03456
Falconer, K.J.: Hausdorff dimension and the exceptional set of projections. Mathematika 29(1), 109–115 (1982)
Falconer, K.J., Howroyd, J.D.: Projection theorems for box and packing dimensions. Math. Proc. Cambridge Philos. Soc. 119(2), 287–295 (1996)
Falconer, K.J., Howroyd, J.D.: Packing dimensions of projections and dimension profiles. Math. Proc. Cambridge Philos. Soc. 121(2), 269–286 (1997)
Fässler, K., Orponen, T.: On restricted families of projections in \({\mathbb{R}}^3\). Proc. Lond. Math. Soc. (3) 109(2), 353–381 (2014)
Federer, H.: The \((\varphi, k)\) rectifiable subsets of \(n\)-space. Trans. Am. Math. Soc. 62, 114–192 (1947)
Fraser, J.M., Orponen, T.: The Assouad dimensions of projections of planar sets. Proc. Lond. Math. Soc. (3) 114(2), 374–398 (2017)
Hovila, R.: Transversality of isotropic projections, unrectifiability, and Heisenberg groups. Rev. Mat. Iberoam. 30(2), 463–476 (2014)
Hovila, R., Järvenpää, E., Järvenpää, M., Ledrappier, F.: Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces. Geom. Dedic. 161, 51–61 (2012)
Iseli, A.: Marstrand-type projection theorems for linear projections and in normed spaces. (2018). arXiv:1809.00636
Iseli, A.: Dimension and projections in normed spaces and Riemannian manifolds. PhD thesis, Universität Bern, Switzerland, 2018. http://biblio.unibe.ch/download/eldiss/18iseli_a.pdf
Kaufman, R.: On Hausdorff dimension of projections. Mathematika 15, 153–155 (1968)
Marstrand, J.M.: Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. 3(4), 257–302 (1954)
Mattila, P.: Hausdorff dimension, projections, intersections, and Besicovitch sets. (2017). arXiv:1712.09199
Mattila, P.: Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Ser. A I Math. 1(2), 227–244 (1975)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability Cambridge Studies in Advanced Mathematics, vol 44. Cambridge University Press, Cambridge (1995)
Mattila, P.: Fourier Analysis and Hausdorff Dimension. Cambridge Studies in Advanced Mathematics, vol. 150. Cambridge University Press, Cambridge (2015)
Orponen, T.: On the dimension and smoothness of radial projections. (2017). arXiv:1710.11053
Orponen, T., Venieri, L.: Improved bounds for restricted families of projections to planes in \({\mathbb{R}}^3\). Int. Math. Res. Not. (2018). https://doi.org/10.1093/imrn/rny193
Peres, Y., Schlag, W.: Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions. Duke Math. J. 102(2), 193–251 (2000)
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The authors would like to thank the referee and the editor for their careful and efficient handling of our paper.
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This research was supported by the Swiss National Science Foundation Grant Nr. 00020 165507.
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Balogh, Z.M., Iseli, A. Projection theorems in hyperbolic space. Arch. Math. 112, 329–336 (2019). https://doi.org/10.1007/s00013-018-1252-3
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DOI: https://doi.org/10.1007/s00013-018-1252-3