SIMPLE RANDOM WALKS ON HIGHER DIMENSIONAL TORI ARE MIXING AND NOT UNIQUELY ERGODIC

The paper’s main theorem and proof strategy are coherent and (modulo minor exposition issues) sound: it sets the EVP–quasi-invariance correspondence correctly via h(z)=p(z)/q(R_α z), uses a dense Gδ set of quasi-invariance equations with unique solutions, builds the non-uniquely-ergodic analytic example in d≥2, and proves atomlessness and mixing under a finite smoothness threshold s(d) (see Theorem 1 and Section 2 of the paper). By contrast, the candidate solution replaces the key correspondence with an incorrect one: it defines p(z)=h(z)/(h(z)+h(z−α)) and then asserts that solving d(µ∘R_α)/dµ = h(z)/h(z−α) yields stationarity for that p, which does not match the paper’s verified condition h=p/(q∘R_α). The candidate also claims a converse (stationary ⇒ quasi-invariant) without the necessary hypotheses acknowledged by the paper (which cites Conze–Guivarc’h for the precise conditions). Hence the model’s argument contains a critical logical error, while the paper’s claims are supported by correct lemmas and known results.