Hausdorff dimension and subgroups of \(\mathrm{SU}(2)\)
type: Item
instance of: scholarly article
title: Hausdorff dimension and subgroups of \(\mathrm{SU}(2)\)
zbMATH Open document ID: 1328.22007
author: Elon Lindenstrauss
author: Nicolas de Saxcé
published in: Israel Journal of Mathematics
publication date: 2015-11-16T00:00:00Z
review text: The Erdős-Volkmann Ring Conjecture, formulated in 1966, stated that there exists no proper Borel measurable subring of \(\mathbb{R}\) which has positive Hausdorff dimension. This conjecture was confirmed by \textit{G. A. Edgar} and \textit{C. Miller} [Proc. Am. Math. Soc. 131, No. 4, 1121--1129 (2003; Zbl 1043.28003)]. In the present paper, an analog of the Erdős-Volkmann Ring conjecture for the group SU(2) is proved: \textit{Any Borel measurable proper dense subgroup of \(\mathrm{SU}(2)\) has Hausdorff dimension zero}. Key ingredients in the proof are the Bourgain-Gamburd Product Theorem in \(\mathrm{SU}(2)\) from [\textit{J. Bourgain} and \textit{A. Gamburd}, Invent. Math. 171, No. 1, 83--121 (2008; Zbl 1135.22010)] and combinatorial arguments such as the Balog-Szemerédi-Gowers Lemma. To obtain the theorem mentioned above, the authors actually prove a more precise quantitative statement, namely a dimensional inequality on product sets.
reviewed by: Christoph Aistleitner
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 43A75
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 6509330
zbMATH Keywords: Erdős-Volkmann Ring Conjecture
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Bourgain-Gamburd Product Theorem
zbMATH Keywords: Balog-Gowers-Szemeredi Theorem
zbMATH Keywords: \(\mathrm{SU}(2)\)
MaRDI profile type: Publication
full work available at URL: s11856-015-1221-5
OpenAlex ID: W2273511180
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DOI: 10.1007/S11856-015-1221-5
description: scientific article; zbMATH DE number 6509330