  In this paper, the available spatial Degrees-Of-Freedoms (DOF) in single
antenna systems is exploited. A new coding scheme is proposed in which several
data streams having fractional multiplexing gains are sent by transmitters and
interfering streams are aligned at receivers. Viewed as a field over rational
numbers, a received signal has infinite fractional DOFs, allowing simultaneous
interference alignment of any finite number of signals at any finite number of
receivers. The coding scheme is backed up by a recent result in the field of
Diophantine approximation, which states that the convergence part of the
Khintchine-Groshev theorem holds for points on non-degenerate manifolds. The
proposed coding scheme is proved to be optimal for three communication
channels, namely the Gaussian Interference Channel (GIC), the uplink channel in
cellular systems, and the $X$ channel. It is proved that the total DOF of the
$K$-user GIC is $\frac{K}{2}$ almost surely, i.e. each user enjoys half of its
maximum DOF. Having $K$ cells and $M$ users within each cell in a cellular
system, the total DOF of the uplink channel is proved to be $\frac{KM}{M+1}$.
Finally, the total DOF of the $X$ channel with $K$ transmitters and $M$
receivers is shown to be $\frac{KM}{K+M-1}$.
