2015/10/20 by Anton Lukyanenko, Lukyanenko, Anton, Joseph Vandehey +1
Mathematics · #11J70 #53C17 #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #Primary 11J83 #Secondary 22E25 #math.MG #math.NT #msc:11J70 #msc:11J83 #msc:22E25 #msc:53C17
paper · pdf · doi:10.48550/arxiv.1510.06033
42 pages
arxiv created 2015/10/20 · arxiv updated 2015/10/22
We initiate the study of an intrinsic notion of Diophantine approximation on a rational Carnot group G. If G has Hausdorff dimension Q, we show that its Diophantine exponent is equal to (Q+1)/Q, generalizing the case G=\mathbb Rn. We furthermore obtain a precise asymptotic on the count of rational approximations. We then focus on the case of the Heisenberg group \mathbb Hn, distinguishing between two notions of Diophantine approximation by rational points in \mathbb Hn: Carnot Diophantine approximation and Siegel Diophantine approximation. After computing the Siegel Diophantine exponent (surprisingly, equal to 1 for all \mathbb Hn), we consider Siegel-badly approximable points to show that Siegel approximation is linked to both Heisenberg continued fractions and to geodesics in complex hyperbolic space. We conclude by showing that Carnot and Siegel approximation are qualitatively different: Siegel-badly approximable points are Schmidt winning in any complete Ahlfors regular subset of \mathbb Hn, while the set of Carnot-badly approximable points does not have this property.