We establish a formal connection between the problem of characterizing degrees of freedom (DoF) in constant single-antenna interference channels (ICs) with general channel matrix and the field of additive combinatorics. The theory we develop is based on a recent breakthrough result by Hochman, 2014, in fractal geometry. Our first main contribution is an explicit condition on the channel matrix to admit full, i.e., K/2 DoF; this condition is satisfied for almost all channel matrices. We also provide a construction of corresponding full DoF-achieving input distributions. The second main result is a new DoF-formula exclusively in terms of Shannon entropy. This formula is more amenable to both analytical statements and numerical evaluations than the DoF-formula by Wu et al., 2015, which is in terms of Rényi information dimension. We then use the new DoF-formula to shed light on the hardness of finding the exact number of DoF in ICs with rational channel coefficients, and to improve the best known bounds on the DoF of a well-studied channel matrix.

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full ie k2 full dofachieving input ics wellstudied channel matrix characterizing degrees of freedom dof rational channel coefficients constant singleantenna interference channels of additive combinatorics rex0301nyi information shannon dofformula

Helmut Bö,David Stotz,lcskei,.Characterizing Degrees of Freedom Through Additive Combinatorics. 62 (11),6423-6435.

Helmut Bö,David Stotz,lcskei,"Characterizing Degrees of Freedom Through Additive Combinatorics" 62

Helmut Bö,David Stotz,lcskei,"Characterizing Degrees of Freedom Through Additive Combinatorics"