Hausdorff dimension and subgroups of \(\mathrm{SU}(2)\)
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.
scientific article; zbMATH DE number 6509330
scientific article