Projection theorems for linear-fractional families of projections

Anton Lukyanenko; Annina Iseli

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arXiv · 2112.12274

Projection theorems for linear-fractional families of projections

Anton Lukyanenko·Annina Iseli

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Abstract

Marstrand's theorem states that applying a generic rotation to a planar set $A$ before projecting it orthogonally to the $x$-axis almost surely gives an image with the maximal possible dimension $\min(1, \dim A)$. We first prove, using the transversality theory of Peres-Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in $PSL(2,\C)$ or a generic real linear-fractional transformation in $PGL(3,\R)$. We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of $PSL(2,\C)$ or $PGL(3,\R)$. Third, we demonstrate, in any dimension, local transversality and resulting projection statements for the families of closest-point projections to totally-geodesic subspaces of hyperbolic and spherical geometries.

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BibTeX RIS

Anton Lukyanenko, Annina Iseli. 2023-06-02. Projection theorems for linear-fractional families of projections. https://arxiv.org/abs/2112.12274

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