SUBMANIFOLD-GENERICITY OF Rd-ACTIONS AND UNIFORM MULTIPLICATIVE DIOPHANTINE APPROXIMATION

The paper’s Theorem 2.1 proves an almost sure bound along sequences tn with gaps tn+1−tn ≥ ζ via an h-th moment estimate (Lemma 3.1) followed by Markov and Borel–Cantelli; the moment bound is obtained by clustering h-tuples into blocks, applying multiple mixing across well-separated blocks, and handling within-block correlations via Lp controls, leading to a main term b^{k(h−⌊h/2⌋)} and a secondary term b^{k(h−1)} together with an exponentially small e^{−bt} remainder. Choosing b = t^{−1+δ/k} yields the desired summable tail and the pointwise rate O(t−k/2+1/h+δ) for ν-a.e. x (Theorem 2.1) . The candidate solution adopts a similar clustering idea and uses exponential mixing on the “far” region and trivial sup-norm bounds on the “near” region, then applies Markov and Borel–Cantelli. However, it incorrectly discards configurations with singleton components by asserting that singletons would place the tuple into the far region (Δ > Rt). This is false: a tuple can have a close pair (Δ ≤ Rt) and still contain singletons far from everyone else. Without treating singletons using mixing across blocks (as the paper does via functions aggregated on cluster anchors), the candidate’s near-region bound is too weak (worst case m = h − 1 gives only t−k (log t)k), which is insufficient for the summability needed for arbitrarily small δ when h ≥ 3. The paper’s proof explicitly handles this via the partition into blocks with #I ≥ 2 and an additional b^{k(h−1)} term, ensuring the sharp moment decay and the final rate . Therefore, the paper is correct and complete, while the model’s solution has a gap.