ERGODICITY OF SKEW PRODUCTS OVER TYPICAL IETS

The paper proves Theorem 1.1 by a careful contradiction scheme that avoids essential-value criteria: it constructs Tf-invariant measures on a compact strip, uses bounded Birkhoff sums along balanced Rauzy–Veech towers (via Zorich acceleration and the KZ spectral gap), and a nudging argument to force translation invariance by each jump on the fiber measures; density of the jump group then forces Lebesgue, yielding a contradiction . The candidate solution instead invokes the essential-values method and claims that each jump is an essential value using Rauzy–Veech towers and ‘partial rigidity,’ but it does not establish the key essential-value criterion (existence, for every B of positive measure, of infinitely many n with μ(B ∩ T^{-n}B ∩ {|S_n f − Δ_j| < ε}) > 0). The presented tower properties are insufficient to guarantee the required B ∩ T^{-n}B intersections; no genuine partial rigidity estimate is proved. Therefore the model’s proof outline is incomplete/incorrect, while the paper’s argument is coherent and complete.