UNIFORM SYNDETICITY IN MULTIPLE RECURRENCE

The paper proves exactly the uniform multiple-recurrence/syndeticity claim the model declared likely open for d ≥ 3. Theorem 1.7 states that for any uniformly amenable group Γ, every d ≥ 1, and any ε>0, there exist δ, η > 0 depending only on ε and d such that BD_Γ({γ: μ(T_1^{γ−1}E ∩ T_2^{γ−1}T_1^{γ−1}E ∩ … ∩ T_d^{γ−1}⋯T_1^{γ−1}E) ≥ δ}) ≥ η; in particular, these return sets are left syndetic . The introduction makes clear the goal is a uniform version of Furstenberg’s multiple recurrence, with Banach-density control and syndeticity, for all d . For Z-actions this is Theorem 1.5, again with δ,η depending only on (ε,d) and implying syndeticity via the Banach-density characterization . The key technical input is a non-conventional ergodic theorem for amenable groups (Theorem 1.9) yielding multiple recurrence and a positive-threshold syndetic set , proved via sated extensions and an uncountable adaptation of Austin’s method . Uniformity in δ,η for a fixed uniformly amenable Γ is obtained by a compactness/ultraproduct argument that uses Keller’s characterization of uniform amenability (ultrapowers remain amenable) and ultralimit lemmas for invariant means and Banach density to contradict any hypothetical failure of uniform bounds . Thus, the model’s claim that the fully uniform result for d ≥ 3 was “likely open as of the cutoff” is refuted by the paper’s main theorem.