Generalized polynomials and mild mixing


An unsettled conjecture of V. Bergelson and I. Håland proposes that if (X, A, μ, T) is an invertible weak mixing measure preserving system, where μ(X) < ∞, and if p1, p2, · · ·, pk are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no pi, nor any pi - pj, i ≠ j, is constant on a set of positive density, then for any measurable sets A0, A1,⋯, Ak, there exists a zero-density set E ⊂ Z such that (Equation Presented) We formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families {p1, p 2, •••, pk} satisfying the hypotheses of this theorem. © Canadian Mathematical Society 2009.

Publication Title

Canadian Journal of Mathematics