Gorodnik, Alexander; Nevo, Amos; Yehoshua, Gal Counting lattice points in norm balls on higher rank simple Lie groups. (English) Zbl 1405.22013 Math. Res. Lett. 24, No. 5, 1285-1306 (2017). Let \(n\geq 2\) and \(\tau: G=\mathrm{SL}_2(\mathbb R)\to \mathrm{GL}_n(\mathbb R)\) an irreducible representation. Then \(\tau\) indices a norm on \(G\), \[ \| x\|_\tau= \sqrt{\mathrm{tr}\tau(X)^t\tau(x)}. \] Let \(\Gamma\subset G\) be a lattice and write \(\kappa= \frac{1}{(n+1)(n+2)}\). For \(R>0\) let \(B^\tau_R\) be the set of all \(x\in G\) with \(\| x\|_\tau\leq R\). By W. Duke et al. [Duke Math. J. 71, No. 1, 143–179 (1993; Zbl 0798.11024)] it is shown that for every \(\varepsilon>0\) one has \[ \frac{| B^\tau_R\cap\Gamma|}{\mathrm{vol}(B^\tau_R)}= \mathrm{vol}(\Gamma\setminus G)+O_\varepsilon\Biggl(\Biggl(\frac{1}{\mathrm{vol}(B^\tau_R)}\Biggr)^{\kappa-\varepsilon}\Biggr),\qquad R\to\infty. \] Using refined spectral estimates based on universal bounds for spherical functions, the authors of the current paper improve the error bound in various ways, in certain cases reaching an improvement of a factor 2, as the following example indicates: For \(\tau\) being the adjoint representation they show that \[ \frac{| B^{\mathrm{Ad}}_R\cap\Gamma|}{\mathrm{vol}(B^{\mathrm{Ad}}_R)}= \text{vol}(\Gamma\setminus G)+ O_\varepsilon \Biggl(\Biggl(\frac{1}{\text{vol}(B^\tau_R)}\Biggr)^{2\frac{n}{n+2}\kappa} \log(R)^q\Biggr),\quad R\to\infty \] for some explicit constant \(q\). Although they only present the case \(G=\mathrm{SL}_n\) the methods apply to any semisimple Lie group. Reviewer: Anton Deitmar (Tübingen) MSC: 22E40 Discrete subgroups of Lie groups 11G99 Arithmetic algebraic geometry (Diophantine geometry) 11P21 Lattice points in specified regions 37A17 Homogeneous flows Keywords:lattice count; Lie group Citations:Zbl 0798.11024 PDFBibTeX XMLCite \textit{A. Gorodnik} et al., Math. Res. Lett. 24, No. 5, 1285--1306 (2017; Zbl 1405.22013) Full Text: DOI arXiv