MEASURABLE TILINGS BY ABELIAN GROUP ACTIONS

The paper’s Theorem 1.2(i)–(ii) is stated and proved rigorously: (i) via a characteristic‑p/Frobenius argument plus a careful de‑collision step and the r = −1 case, yielding F^r ⋄ A when r is coprime to all primes ≤ |F|; (ii) via the mean ergodic theorem to obtain f^q‑invariant limit functions with the stated properties . The candidate solution asserts the correct conclusions but (i) replaces the key Frobenius/convolution proof with an unsubstantiated “mass-transport”/cycle argument, and (ii) constructs Cesàro limits without invoking the mean ergodic theorem or a valid substitute; as written, the limits and invariance are not justified. Hence the paper is correct, while the model’s proof is incomplete/incorrect.