Explicit and almost sure conditions for K/2 degrees of freedom

Abstract: It is well known that in K-user constant single-antenna interference channels K/2 degrees of freedom (DoF) can be achieved for almost all channel matrices. Explicit conditions on the channel matrix to admit K/2 DoF are, however, not available. The purpose of this paper is to identify such explicit conditions, which are satisfied for almost all channel matrices. We also provide a construction of corresponding asymptotically DoF-optimal input distributions. The main technical tool used is a recent breakthrough result by Hochman in fractal geometry.