  The \emph{Diophantine exponent} of an action of a group on a homogeneous
space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of
approximating the points of the homogeneous space by the points on an orbit of
the group. We show that the Diophantine exponent of the
$\mathrm{SL}_n(\mathbb{Z}[1/p])$-action on the generalized upper half-space
$\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}_n(\mathbb{R})$, lies in $[1,1+O(1/n)]$,
substantially improving upon Ghosh--Gorodnik--Nevo's method which gives the
above range to be $[1,n-1]$. We also show that the exponent is \emph{optimal},
i.e.\ equals one, under the assumption of \emph{Sarnak's density hypothesis}.
The result, in particular, shows that the optimality of Diophantine exponents
can be obtained even when the \emph{temperedness} of the underlying
representations, the crucial assumption in Ghosh--Gorodnik--Nevo's work, is not
satisfied. The proof uses the spectral decomposition of the homogeneous space
and bounds on the local $L^2$-norms of the Eisenstein series.
