Number Theory Meets Wireless Communications: An Introduction for Dummies Like Us

The paper rigorously proves DoF = 4/3 for the constant real 2×2 X-channel for almost every set of channel coefficients via a Khintchine–Groshev (KG) lower bound on minimum distance and further strengthens this to hold outside a Hausdorff-dimension ≤ 4 − 1/3 exceptional set using jointly singular sets (Theorems 6 and 10) . The candidate solution reproduces the same alignment mapping and correctly recovers the a.e. DoF argument, but its exceptional-set dimension argument is flawed: it attempts to bound the bad set by a “very well approximable” (Wv) class via Jarník/MTP, which does not contain all failures of the logarithmically strengthened KG bound, and then (incorrectly) passes this bound to a superset. It also treats the inversion map α → (α1−1, α2−1) as preserving the specific Diophantine property needed, which is not justified. Thus the model’s proof of the Hausdorff-dimension claim is invalid, even though the numerical conclusion matches the paper’s (which is established by a different method via jointly singular sets).