The Furstenberg–Sárközy Theorem and Asymptotic Total Ergodicity Phenomena in Modular Rings

The paper’s Theorem 2.6 states exactly the target statement and gives a complete proof via van der Corput differencing plus a quantitative bound that goes to zero as lpf(Nm)→∞ . The candidate solution proves the same theorem by a different route: Fourier diagonalization, complete exponential sums modulo N, CRT factorization, and a sharp local prime-power bound using a derivative test. That argument is essentially sound and yields an explicit operator-norm bound O((deg P−1)/lpf(N)). One minor flaw is the claim that for any nonzero j mod N there exists p|N with p∤j; this is false in general (e.g., N=12, j=6), but the proof only needs a prime-power p^a||N with vp(j)<a, which always exists for j∈{1,…,N−1}. With that fix, the model’s proof aligns with the paper’s result. The paper’s derivation (Lemma 2.4 → (2.44)) contains what appears to be a typographical slip in the displayed exponent in (2.44), but the surrounding argument clearly implies a bound tending to 0 as lpf(N)→∞ .