The Degrees-of-Freedom of the K-User Gaussian Interference Channel Is Discontinuous at Rational Channel Coefficients

The degrees-of-freedom of aK-user Gaussian interference channel (GIC) has been defined to be the multiple of(1/2)log2Pat which the maximum sum of achievable rates grows with increasing powerP. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GICs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are nonzero rational numbers. More specifically, for allK> 2, we find a class ofK-user GICs that is dense in the GIC parameter space for whichK/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIC with nonzero rational coefficients is strictly smaller thanK/2. These results are proved using new connections with number theory and additive combinatorics.