  In this paper, we establish Cwikel-type estimates for noncommutative tori for
any dimension~$n\geq 2$. We use them to derive Cwikel-Lieb-Rozenblum
inequalities and and Lieb-Thirring inequalities for the number of negative
eigenvalues of fractional Schroedinger operators on noncommutative tori in any
dimension~$n\geq 2$. The latter leads to a Sobolev inequality for
noncommutative tori. On the way we establish a "borderline version" of the
abstract Birman-Schwinger principle for the number of negative eigenvalues of
relatively compact form perturbations of a non-negative semi-bounded operator
with isolated 0-eigenvalue.
