Directional Ergodicity and Weak Mixing for Actions of Rd and Zd

The paper proves the Td wall-criterion for directional ergodicity/weak mixing of Zd-actions via the unit suspension and the aperiodic spectral measure, reducing to the Rd case and then pulling back/pushing forward measures (notably Lemma 4.6 and Theorem 4.8 leading to Theorem 4.10) . The model independently derives the same criterion by a fiberwise spectral transform on L2_O(X̃), a gauge that removes the suspension cocycle, and a Fourier expansion in the torus variable, yielding precisely the affine coset condition P = π(L⊥)+π(ℓ). The only adjustment needed in the model write-up is to state the paper’s ergodicity hypothesis for T explicitly (as in Theorem 4.10); otherwise, the logic matches the paper’s result and is correct.