Projection theorems for linear-fractional families of projections

TL;DR

This paper extends Marstrand's projection theorem to complex and real linear-fractional transformations, showing that generic transformations preserve maximal projection dimension under certain conditions.

Contribution

It proves that Marstrand's theorem applies to a broader class of transformations, including complex and real linear-fractional maps, and explores transversality for specific subgroup families.

Findings

Maximal projection dimension preserved under generic complex linear-fractional transformations.

Transversality holds for certain subgroup families of transformations.

Projection results extend to hyperbolic and spherical geometries.

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.