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The degrees-of-freedom of a K-user Gaussian interference channel (GIFC) hasbeen defined to be the multiple of (1/2)log_2(P) at which the maximum sum ofachievable rates grows with increasing P. In this paper, we establish that thedegrees-of-freedom of three or more user, real, scalar GIFCs, viewed as afunction of the channel coefficients, is discontinuous at points where all ofthe coefficients are non-zero rational numbers. More specifically, for all K>2,we find a class of K-user GIFCs that is dense in the GIFC parameter space forwhich K/2 degrees-of-freedom are exactly achievable, and we show that thedegrees-of-freedom for any GIFC with non-zero rational coefficients is strictlysmaller than K/2. These results are proved using new connections with numbertheory and additive combinatorics.

On the Degrees-of-Freedom of the K-User Gaussian Interference Channel