The classical notion of a Diophantine exponent, \(\kappa\), can be extended and generalised in many ways. In recent work,A. Ghosh et al. [J. Reine Angew. Math. 745, 155–188 (2018;Zbl 1405.37012)] considered the very general setting of a lattice \(\Gamma\) in a connected Lie (or algebraic) group \(G\), acting on a homogeneous space \(G/H\), where \(H\) is a subgroup of \(G\). When a condition called temperedness is satisfied, they were able to obtain the optimal Diophantine exponent, \(\kappa=1\).
Here the authors consider the important case when \(\Gamma=\mathrm{SL}_{n} \left( {\mathbb Z}[1/p] \right)\), \(G=\mathrm{SL}_{n} \left( {\mathbb R} \right) \times \mathrm{SL}_{n} \left( {\mathbb Q}_{p} \right)\) and \(H=\mathrm{SO}_{n} \left( {\mathbb R} \right) \times \mathrm{SL}_{n} \left( {\mathbb Q}_{p} \right)\). As in the classical setting, where \({\mathbb Q}\) is dense in \({\mathbb R}\), it is known that \(\mathrm{SL}_{n} \left( {\mathbb Z}[1/p] \right)\) is dense in \(\mathrm{SL}_{n} \left( {\mathbb R} \right)\), so this is a natural setting for investigating Diophantine exponents. However, the temperedness assumption does not hold here. The work of Ghosh, Gorodnik and Nevo does provide upper bounds for \(\kappa\) (e.g., \(\kappa \leq n-1\) for \(n \geq 3\)) here, but these bounds are not optimal. In this work, the authors improve these upper bounds, obtaining bounds that approach \(1\) as \(n\) increases.
In Theorem 2, they obtain the optimal Diophantine exponent for \(n=2\) and \(3\); i.e., they show that \(\kappa=1\) for such \(n\). For \(n \geq 4\), they obtain an upper bound for \(\kappa\) that depends on bounds for the Generalised Ramanujan Conjecture (GRC) for \(\mathrm{GL}(n)\) (see Section 4.4 of this paper for its definition, as well as for Sarnak’s Density Hypothesis below). Given the best proven bounds that we currently have for GRC, they show that \(\kappa \leq 11/8\) for \(n=4\) and that \(\kappa \leq \left( n^{2}+1 \right)/ \left( n^{2}-n \right) =1 + O(1/n)\) for \(n \geq 5\). So as \(n\) grows, they obtain results increasingly close to the optimal value of \(1\).
Furthermore, they are able to prove that \(\kappa=1\) under Sarnak’s Density Hypothesis for \(\mathrm{GL}(n)\) (see their Theorem 3). This is of interest since this hypothesis is much weaker than GRC. This answers negatively a question that Ghosh, Gorodnik and Nevo ask in Remark 3.6 of their paper cited above: if \(\kappa=1\), does it follow that the temperedness assumption holds?
Finally, recall that a Diophantine exponent, \(\kappa \left( x_{0} \right)\), can be defined for individual elements too (\(\kappa\) is an infimum obtained from them). Due to local obstructions, this may be larger than \(\kappa\) for some elements, \(x_{0}\). The authors conjecture that \(\kappa \left( x_{0} \right)=1\) always holds in their setting. This is known, under GRC, for \(n=2\). Here, in their Theorem 4, the authors also prove that this is true for \(n=3\), subject to GRC.

Reviewer: Paul Voutier (London)

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