ON DIVERGENT ON AVERAGE TRAJECTORIES FOR HIGHER RANK ACTIONS

Kim’s paper proves Theorem 1.3 that dim_H{x in UΓ: x is A′-divergent on average} ≤ (d−1)/2, via one-parameter dynamical height functions β with average-contraction (eq. (1.7)) and a higher-rank ψ built from them, together with an independence/tensorization step (Prop. 3.4) and a covering estimate (Prop. 4.1), culminating in the stated dimension bound (end of §4.1) . The candidate solution follows this same blueprint at a high level: identify UΓ with the torus, use β-type height functions contracting along each A_i, tensorize to exploit independence, and pass from integral bounds to coverings to get the (d−1)/2 bound. Minor inaccuracies include misstating the size of the A′-mesh (polynomial, not exponential in N) and integrating over U rather than individual U_i directions, but these do not alter the core argument’s logic.