  Let $\Sigma_d^{++}$ be the set of positive definite matrices with determinant
1 in dimension $d\ge 2$. Identifying any two $SL_d(\mathbb{Z})$-congruent
elements in $\Sigma_d^{++}$ gives rise to the space of reduced quadratic forms
of determinant one, which in turn can be identified with the locally symmetric
space $X_d:=SL_d(\mathbb{Z})\backslash SL_d(\mathbb{R})/SO_d(\mathbb{R})$.
Equip the latter space with its natural probability measure coming from a Haar
measure on $SL_d(\mathbb{R})$. In 1998, Kleinbock and Margulis established
sharp estimates for the probability that an element of $X_d$ takes a value less
than a given real number $\delta>0$ over the non--zero lattice points
$\mathbb{Z}^d\backslash\{ 0 \}$.
  In this article, these estimates are extended to a large class of probability
measures arising either from the spectral or the Cholesky decomposition of an
element of $\Sigma_d^{++}$. The sharpness of the bounds thus obtained are also
established (up to multiplicative constants) for a subclass of these measures.
  Although of an independent interest, this theory is partly developed here
with a view towards application to Information Theory. More precisely, after
providing a concise introduction to this topic fitted to our needs, we lay the
theoretical foundations of the study of some manifolds frequently appearing in
the theory of Signal Processing. This is then applied to the recently
introduced Integer-Forcing Receiver Architecture channel whose importance stems
from its expected high performance. Here, we give sharp estimates for the
probabilistic distribution of the so-called \emph{Effective Signal--to--Noise
Ratio}, which is an essential quantity in the evaluation of the performance of
this model.
