Entropic van der Corput’s Difference Theorem

The paper proves Æ(x) = Æ(x^d) for sequences x: N → R/Z by (i) showing Æ(x) ≤ Æ(x^d) via a constructive approximation-and-block-counting Lemma 3.1, together with lower semicontinuity (Lemma 2.5), and (ii) showing Æ(x^d) ≤ Æ(x) using that x(n+d) − x(n) lies in the anqie generated by x, hence the anqie of the difference is a subanqie with no larger entropy (Lemma 2.3(ii)); see Theorem 1.1 and its proof and preliminaries . The candidate solution proves the same equality by a different, fully topological route: it identifies the anqie system with the subshift orbit-closure model (consistent with the paper’s equivalence of constructions ), observes that the difference map is a sliding-block factor (hence Æ(x^d) ≤ Æ(x)), and then embeds X(x) as a conjugate subsystem of an isometric skew product over X(x^d), whose topological entropy equals that of the base; this yields Æ(x) ≤ Æ(x^d). Both arguments are logically sound and arrive at the same result; they are methodologically distinct (paper: C*-algebra/approximation; model: factor/extension in topological dynamics).