Characterizing degrees of freedom through additive combinatorics

TL;DR

This paper links the characterization of degrees of freedom in interference channels to additive combinatorics, providing explicit conditions for full DoF, a new entropy-based DoF formula, and insights into the complexity of computing DoF.

Contribution

It introduces a novel connection between DoF in interference channels and additive combinatorics, offering explicit conditions, a new entropy-based formula, and improved bounds.

Findings

Explicit condition for full DoF in almost all channel matrices

A new DoF formula based on Shannon entropies

Enhanced bounds on DoF for specific channel matrices

Abstract

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 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 DoF-optimal input distributions. The second main result is a new DoF-formula exclusively in terms of Shannon entropies. This formula is more amenable to both analytical statements and numerical evaluations than the DoF-formula by Wu et al., which is in terms of R\'enyi 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.