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

Abstract

The degrees-of-freedom of a <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -user Gaussian interference channel (GIC) has been defined to be the multiple of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(1/2)log</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> at which the maximum sum of achievable rates grows with increasing power <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</i> . 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 all <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> > 2, we find a class of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -user GICs that is dense in the GIC parameter space for which <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> /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 than <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> /2. These results are proved using new connections with number theory and additive combinatorics.