IP-systems, generalized polynomials and recurrence


Let Ω be an abelian group. A set R ⊂ Ω is a set of recurrence if, for any probability measure-preserving system (X, B, μ, {Tg}g∈Ω) and any A ∈ A with μ(A) > 0, μ(A ∩ TgA) > 0 for some g ∈ R. If (x i)i=1∞ is a sequence in Ω2, the set of its finite sums {xi1 + xi2 + ⋯ + x ik: i1 < i2 < ⋯ < i k} is called an IP-set. In Bergelson et al (Erg. Theor. Dyn. Sys. 16 (1996), 963-974) it is shown that if p: ℤd→ℤ k is a polynomial vanishing at zero and F is an IP-set in ℤd then (p(n): n ∈ F] is a set of recurrence in ℤk. Here we extend this result to an analogous family of generalized polynomials, that is functions formed from regular polynomials by iterated use of the greatest integer function, as a consequence of a theorem establishing a much wider class of recurrence sets occurring in any (possibly non-fmitely generated) abelian group. While these sets do in a sense have a distinctively 'polynomial' nature, this far-ranging class includes, even in ℤ, such examples as {∑i,j∈α,i

Publication Title

Ergodic Theory and Dynamical Systems