Fig. 2. The two-user X channel. received direction (coefficient). To show this in fact reduc es the total dimension of the received interference, we consider the two-user X channel. In the two-user X channel, each transmitter has independent messages to both receivers, see Figure 2. Hence, each transmitter has two data streams and they need to be transmitted such that they can be separated in their corresponding receivers. In [8], the following signaling is proposed for the channel: where u$_{1}$, u$_{2}$ and v$_{1}$, v$_{2}$ are data streams intended for the first and second receivers, respectively. All data streams a re transmitted using the constellation U = A ( − Q, Q )$_{Z}$ , where Q is an integer and A is the factor controlling the minimum distance of the received constellation. Using the above signaling, the received signal can be written as: for all p ∈ Z and q ∈ Z $^{m}$. Using the theorem, one can obtain d$_{min}$ ≈ A Q 2 where d$_{min}$ is the minimum distance in the received constellation. It can be shown that if d$_{min}$ ≈ 1 then the additive gaussian noise can be removed from the received signal using an appropriate coding scheme [13]. This condition can be enforced by A ≈ Q $^{2}$. In a noise-free environment, the receiver can decode the three messages if there is a one-to-one map from the received signal to the transmit constellation. This condition can be satisfi ed if a and b are rationally independent which in fact holds almost surely. Therefore the receiver can decode all three messages almost surely. has measure zero for p ∈ $^{Z}$, q ∈ Z $^{m}$, and Q = max {| q$_{1}$ | , . . . , | q$_{m}$ |} . Put it differently, the theorem states that for almost all v ∈ R $^{m}$, there is a constant κ which is only related to v such that The received signals are linear combinations of three terms in which two of them are the intended data streams and one is the sum of interfering signals, see Figure 2. Let us focus on the first receiver. y$_{1}$ resembles the received signal of a multiple access channel with three users. However, there is an important difference between them. In the two-user X channel the term corresponding to the interfering signals, i.e. u$_{3}$ = v$_{1}$ + v$_{2}$ , is a sum of two data streams. However, we claim that this difference does not change considerably the minimum distance of the received constellation, i.e. d$_{min}$ . Recall that Khintchine- Groshev theorem is used to bound d$_{min}$ . The bound is a function of the maximum value that the integers can take. The maximum value of u$_{3}$ is 2 AQ , which is different from a single data stream by a factor of two. Since this change only affects the constant term of Khintchine-Groshev theorem, we have d$_{min}$ ≈ A Q 2 and the receiver can decode all data streams if each of them have a multiplexing gain of 1 $_{3}$. Therefore, the multiplexing gain of 4 3 is achievable in total, which meets the upper bound. where r$_{i}$ is the achievable DOF for User i . In the preceding example, we implicitly assumed that the pair ( a, b ) can take any value in R $^{2}$. Otherwise, we were not able to apply the Khintchine-Groshev theorem. Let us assume that a and b have a relation. For instance, b is a function of a , say b = a $^{2}$. In this case, the pair ( a, b ) lies on a one dimensional manifold in R $^{2}$, see Figure 1. Since the manifold itself has measure zero, Khintchine-Groshev theorem can not be applied directly. For such cases, however, there is an extension to Khintchine-Groshev theorem , see [11] and [12], which states that the same lower bound on the minimum distance can be applied when coefficients lie on a non-degenerate manifold and, in fact, the measure of points not satisfying the theorem is zero. In can be shown that if all v$_{i}$ ’s in (1) are monomials with variables from the set g = { g$_{1}$, g$_{2}$, . . . , g$_{n}$ } , then v lines on a non-degenerate manifold. As a special case when set v has only one member, i.e. v = { 1 , g, g $^{2}$, g$^{3}$, . . . } . In real interference alignment we say two data streams are aligned at a receiver if they arrive at the receiver with the same The signaling proposed for the two-user X channel can be interpreted as follows. The received signal at each receiver is a real number, which is a one-dimensional component. One can embed three rational dimensions, each of which has dimension 1 3 in this one dimensional space, see Figure 2. One of these dimensions is associated with interference and the other two with intended signals. Therefore, 4 3 out of two dimensions To calculate User i ’s rate R$_{i}$ = log(2 Q − 1) in terms of P , we need to find a relation between Q and P . Due to the power constraints, we have P = A $^{2}$Q$^{2}$. We showed that A ≈ Q $^{2}$. Therefore, P ≈ Q $^{6}$. Hence, we have 2 Fig. 1. There are infinitely many points on the a − b plane with measure zero that are not addressed by the Khintchine-Groshev theorem (these are called bad events). The curve b$_{=}$ a 2 is a non-degenerate manifold and by the extension of Khintchine-Groshev theorem, the measure of bad events is zero on the curve. 1 1 a b Bad Events Non-degenerate Manifold b = a 2 x$_{1}$ x$_{2}$ y$_{1}$ y$_{2}$ $^{h}$11 h 1 2 h 2 1 $^{h}$22 v$_{2}$ u$_{2}$ v$_{1}$ u$_{1}$ v$_{2}$ u$_{2}$ v$_{1}$ u$_{1}$ 1 3 1 3 1 3 1 3 1 3 1 3 v$_{1}$ v$_{2}$ u$_{1}$ u$_{2}$ | p + v · q | > κ Q m + ϵ y$_{1}$ = ( h$_{11}$h$_{22}$ ) u$_{1}$ + ( h$_{21}$h$_{12}$ ) u$_{2}$ + ( h$_{11}$h$_{12}$ )( v$_{1}$ + v$_{2}$ ) + z$_{1}$, y$_{2}$ = ( h$_{21}$h$_{12}$ ) v$_{1}$ + ( h$_{11}$h$_{22}$ ) v$_{2}$ + ( h$_{21}$h$_{22}$ )( u$_{1}$ + u$_{2}$ ) + z$_{2}$. x$_{1}$ = h$_{22}$u$_{1}$ + h$_{12}$v$_{1}$, x$_{2}$ = h$_{21}$u$_{2}$ + h$_{11}$v$_{2}$, r$_{i}$ = lim P →∞ R$_{i}$ 0 . 5 log P = 1 3 , (2)