Note on Polynomial Recurrence

Let (X,B, μ, T ) be a measure-preserving system. The well-known Poincare recurrence theorem asserts for any A ∈ B with μ(A) > 0, there exist infinitely many n ∈ N such that μ(A ∩ TA) > 0. For a subset S of N, we say S is syndetic if there exists a constant C > 0 such that S ∩ [x, x + C] 6= ∅ for any sufficiently large x, i.e., S has bound gaps. In [5], Khintchine proved that for any A ∈ B with μ(A) > 0 and any ǫ > 0, the set