Idempotent ultrafilters, multipleweak mixing and Szemerédi’s theorem for generalized polynomials

Abstract

It is possible to formulate the polynomial Szemerédi theorem as follows: Let q _i (x) ∈ Q[x] with q _i (Z) ⊂ Z, 1 ≤ i ≤ k. If E ⊂ N has positive upper density, then there are a, n ∈ N such that

$$ \{ a,a + q_1 (n) - q_1 (0),...,a + q_k (n) - q_k (0)\} \subset E. $$

Using methods of abstract ergodic theory, topological algebra in β N, and some recently obtained knowledge concerning the relationship between translations on nilmanifolds and the distribution of bounded generalized polynomials, we prove, among other results, the following extension, valid for generalized polynomials (functions obtained from regular polynomials via iterated use of the floor function). Let q _i (x) be generalized polynomials, 1 ≤ i ≤ k, and let p ∈ β N be an idempotent ultrafilter all of whose members have positive upper Banach density. Then there exist constants c _i , 1 ≤ i ≤ k, such that if E ⊂ Z has positive upper Banach density, then the set {n ∈ N: ∃ a ∈ Z with {a, a+q ₁(n)−c ₁, …, a+q _k (n)−c _k } ⊂ E} belongs to p. As part of the proof, we also obtain a new ultrafilter polynomial ergodic theorem characterizing weak mixing.