EQUIDISTRIBUTION OF AFFINE RANDOM WALKS ON SOME NILMANIFOLDS

The paper proves the reduction from X to Y under τZ(µ) < 2σX,Y(µ) via a careful Cauchy–Schwarz-based argument (Lemma 4.5) and a quantitative projection estimate (Proposition 4.1), then lifts the torus obstruction using a height-stability lemma to conclude (λ′,α)-quantitative equidistribution on X from that on Y (Theorem 1.1) . The candidate solution’s Step 2 asserts a uniform pointwise decay bound supx |U(µ)m f(x)| ≲ exp(−(σ − τ/2 − ε)m) for the fiber-complement (E(f|Y)=0), derived from L2 contraction and a counting bound; this L2→C0 conversion is not justified because the argument controls only averaged (L2) norms, not the sup over the base variable. The paper explicitly avoids this gap by constructing a fiber-constant witness and working with Wasserstein via integration against η (rather than pointwise sup), and by inserting a short time-splitting m′ to retain quantitative mass on Y before applying equidistribution there and lifting back to X . Hence the paper’s proof is correct and complete, while the model’s proof outline contains a key unjustified step.