A well-known result of V. Bergelson [Ergodic Theory Dyn. Syst. 7, 337–349 (1987; Zbl 0645.28012)] states that if \((X,\mathcal A , \mu ,T)\) is an invertible weakly mixing system, and the polynomials \(p_1,\dots , p_k\in \mathbb Z [t]\) have the property that no \(p_i\) or \(p_i-p_j\) is constant for \(1\leq i\neq j\leq k\), then for any measurable sets \(A_0,A_1,\dots , A_k\) there exists a zero density set \(E\) such that \[ \lim_{n\to\infty, n\notin E} \mu(A_0\cap T^{p_1(n)}(A_1)\cap\dots \cap T^{p_k(n)}(A_k))=\prod_{i=0}^k\mu(A_i). \] An interesting next question that arises is this: do polynomial functions constitute a suitably “most general class” of integer sequences along which one can guarantee “weak mixing of all orders”? Bergelson and Håland conjectured that the class of standard polynomials in the previous statement can be extended to the class of generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function).
The authors formulate and prove a faithful version of this conjecture for mildly mixing systems. Moreover, they partially characterize the set of families of generalized polynomials satisfying the hypothesis of their main result.