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are aligned and the number of effective received directions is L ′ 2 = n + 1 . Clearly, C2 and C3 hold for Receiver 3. 

Since C1, C2, and C3 hold at all users, we only need to obtain the number of maximum received directions at all receivers. To this end, we observe that 

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. Therefore, an application of Theorem 3 reveals that the following DOF is achievable. 

Since n is an arbitrary integer, one can conclude that 3 2 is achievable for the three-user GIC almost surely. 

C. K -user Gaussian Interference Channel: DOF = K 2 is Achievable 

To prove the main result of the paper, we start with selecting the transmit directions for User i . A direction T ∈ G ( h ) is chosen as the transmit direction for User i if it can be represented as K K 

where s$_{jl}$ ’s are integers satisfying 

   $^{s$_{jj}$}$ = 0 ∀ j ∈ { 1 , 2 , . . ., K } 0 ≤ s$_{ji}$ ≤ n − 1 ∀ j ∈ { 1 , 2 , . . ., K } & j ̸ = i 0 ≤ s$_{jl}$ ≤ n Otherwise. T$_{i}$ 

The set of all transmit directions is denoted by . It is easy to show that the cardinality of this set is 

Clearly, T$_{i}$ satisfies C1 for all i ∈ { 1 , 2 , . . ., K } . ′ 

To compute L i (the number of independent received di- rections due to interference), we investigate the effect of Transmitter k on Receiver i . Let us first define T$_{r}$ as the set of directions represented by (12) and satisfying 

We claim that T$_{ik}$ , the set of received directions at Receiver i due to Transmitter k , is a subset of T$_{r}$ . In fact, all transmit directions of Transmitter k arrive at Receiver i multiplied by h$_{ik}$ . Based on the selection of transmit directions, however, the maximum power of h$_{ik}$ in all members of T$_{ik}$ is n − 1 . There- fore, none of the received directions violates the condition of (14) and this proves the claim. 

Since T$_{r}$ is not related to User k , one can conclude that T$_{ik}$ ⊆ T$_{r}$ for all k ∈ { 1 , 2 , . . ., K } and k ̸ = i . Hence, we deduce that all interfering users are aligned in the directions of T$_{r}$ . Now, L ′ i can be obtained by counting the members of T$_{r}$ . It is easy to show that 

{ s$_{jj}$ = 0 ∀ j ∈ { 1 , 2 , . . ., K } 0 ≤ s$_{jl}$ ≤ n Otherwise. (14) 

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The received directions at Receiver i are members of h$_{ii}$ T$_{i}$ and T$_{r}$ . Since h$_{ii}$ does not appear in members of T$_{r}$ , the members of h$_{ii}$ T$_{i}$ and T$_{r}$ are distinct. Therefore, C2 holds at Receiver i . Since all the received directions are irrationals, C3 does not hold at Receiver i . 

Since C$_{1}$ and C$_{2}$ hold for all users, we can apply Theorem 3 to obtain the DOF of the channel. We have 

where m is 

Combining the two equations, we obtain 

Since n can be arbitrary large, we conclude that K 2 is achiev- able for the K -user GIC. 

[1] M. A. Maddah-Ali, A. S. Motahari, and A. K. Khandani, “Com muni- cation over MIMO $^{X}$channels: Interference alignment, decomposition, and performance analysis,” Information Theory, IEEE Transactions on , vol. 54, no. 8, pp. 3457–3470, August 2008. 

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[8] ——, “Real interference alignment with real numbers,” http://arxiv.org/abs/0908.1208 , August 2009. 

[9] A. Ghasemi, A.S. Motahari, and A.K. Khandani, “Interfer ence Alignment for the K User MIMO Interference Channel,” http://arxiv.org/abs/0909.4604 , 2009. 

[10] M.A. Maddah-Ali, “On the Degrees of Freedom of the Compound MIMO Broadcast Channels with Finite States,” http://arxiv.org/abs/0909.5006 , September 2009. 

[11] V. Bernik, D. Kleinbock, and G. Margulis, “Khintchine- type theorems on manifolds: the convergence case for standard and multiplicative versions,” International Mathematics Research Notices , no. 9, p. 453486, 2001. 

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[13] A. S. Motahari, S. O. Gharan, M.A. Maddah-Ali, and A. K. Khan- dani,“Real Interference Alignment: Exploiting the Potent ial of Single Antenna Systems,” http://arxiv.org/abs/0908.2282 

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