{"title":"Projection theorems for linear-fractional families of projections","authors":"Annina Iseli, Anton Lukyanenko","doi":"10.1017/S0305004123000373","DOIUrl":null,"url":null,"abstract":"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 \n$\\min(1, \\dim A)$\n . 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 \n$PSL(2,\\mathbb{C})$\n or a generic real linear-fractional transformation in \n$PGL(3,\\mathbb{R})$\n . We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of \n$PSL(2,\\mathbb{C})$\n or \n$PGL(3,\\mathbb{R})$\n . 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.","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":"25 1","pages":"625 - 647"},"PeriodicalIF":0.6000,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Proceedings of the Cambridge Philosophical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S0305004123000373","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
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
$PSL(2,\mathbb{C})$
or a generic real linear-fractional transformation in
$PGL(3,\mathbb{R})$
. We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of
$PSL(2,\mathbb{C})$
or
$PGL(3,\mathbb{R})$
. 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.
期刊介绍:
Papers which advance knowledge of mathematics, either pure or applied, will be considered by the Editorial Committee. The work must be original and not submitted to another journal.