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Generalized Degrees of Freedom of the Symmetric Gaussian $K$ User Interference Channel

We characterize the generalized degrees of freedom of the $K$ user symmetric Gaussian interference channel where all desired links have the same signal-to-noise ratio (SNR) and all undesired links carrying interference have the same interference-to-noise ratio, ${INR}={SNR}^α$. We find that the number of generalized degrees of freedom per user, $d(α)$, does not depend on the number of users, so that the characterization is identical to the 2 user interference channel with the exception of a singularity at $α=1$ where $d(1)=\frac{1}{K}$. The achievable schemes use multilevel coding with a nested lattice structure that opens the possibility that the sum of interfering signals can be decoded at a receiver even though the messages carried by the interfering signals are not decodable.