Ergodic Theorems for Bilinear Averages, Roth’s Theorem and Corners Along Fractional Powers

The paper proves the L2-equality of bilinear averages along ⌊h(n)⌋ and along n for c ∈ [1,23/22) via a quantitative change-of-variables method and U3-Gowers norm bounds on a carefully constructed error kernel. The candidate solution’s key step—reducing the bilinear averages to a direct integral of linear averages via the spectral theorem for S—is not mathematically valid: the projection-valued spectral integral cannot be applied to a vector-valued integrand (ζ ↦ (1/N)∑(ζU_T)^{a_n}f) that does not commute with the spectral measure of U_S, so the proposed representation of the bilinear average is unfounded. Even granting the (largely standard) exponential-sum decay used for a Blum–Hanson condition, it only controls linear averages for a single unitary and does not by itself justify equality for bilinear products. The paper’s proof and range are sound; the model’s proof is unsupported at its core reduction step.