Dimension Drop for Diagonalizable Flows on Homogeneous Spaces

The paper proves a uniform codimension drop along unstable leaves under (EEP)+(ENDP) via a careful two-pronged covering argument: (i) effective equidistribution on compact (thick) parts (Definition 1.1 and Proposition 4.4) and (ii) effective non-divergence in the cusp via height functions and iterated Margulis inequalities (Sections 6–7). These are combined in Theorem 2.1 to yield Theorem 1.5 (uniform positive codimension for {h in P : h x ∈ Ẽ(F,O)}), and then sliced to obtain the global result on X (Corollary 1.6) . The candidate solution sketches a porosity-based proof that mirrors the high-level plan (EEP to get many hits into O; ENDP to keep a thick subset; then a Cantor-type deletion to force porosity). However, it makes a critical uniformity error: it applies Chebyshev/Markov to the ENDP inequality to claim a uniform-in-x lower bound on the thick proportion inside every leaf-ball at each time t (“choose R so large that sup_{t,x} C_reg(c0 u_t(x)+d_t)/R < κ/2”). This is impossible because u_t(x) is unbounded in x; ENDP controls averages in terms of u_t(x) and d_t, not uniformly over all x. In the paper, this obstruction is handled by a more delicate covering argument for excursions to infinity (Proposition 6.1 and the coverings of §7), then combined with the EEP-driven coverings via Proposition 8.1 to obtain codimension bounds without ever needing a uniform thick proportion for all x and all times . The candidate also omits the explicit r0(x)-dependent mixing-time threshold in (EEP) (t ≥ a + b log(1/r0(x))) that the paper carefully tracks (Definition 1.1), and uses it improperly as if a single large t worked uniformly for all x . Because these issues drive the central “hole at every scale” step, the porosity conclusion (and thus the dimension bound) is not justified as stated.