POINTWISE ERGODIC THEOREMS FOR NON-CONVENTIONAL BILINEAR AVERAGES ALONG (⌊n^c⌋, −⌊n^c⌋)

The paper proves almost-everywhere convergence of the bilinear averages along (⌊h(n)⌋, −⌊h(n)⌋) for h in the c-regularly varying class with c in [1, 23/22) and further identifies the limit with Bourgain’s bilinear limit, see Theorem 1.7 and equation (1.9) in the uploaded PDF . Its strategy splits the averages via the inverse ϕ = h^{-1} into a main Toeplitz-weighted term that matches Bourgain’s limit (using summation by parts/Toeplitz) and an error controlled by a truncated Fourier expansion of the sawtooth Φ plus Gowers U3-norm bounds and transference; see the Strategy subsection and Section 2–4 . The candidate solution follows the same blueprint: identical change-of-variables and Toeplitz reduction for the main term; the same sawtooth truncation, oscillatory kernel analysis, and Gowers-uniformity/transference for the error. Minor deviations are bibliographic (it cites Demeter instead of Bourgain for the basic bilinear limit) and in technical presentation (L2 versus the paper’s L1-lacunary summability), but the logical path and conclusions coincide.