  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.
