Multiple return times theorems for weakly mixing systems

We prove the pointwise convergence of weighted averages 1/N ΣN n=1ang(Rnz) where (Z, K, v, R ) is an ergodic dynamical system. The sequence an is given by expression of the form an = an(x, y1, y2, ..., yj)=(HΠi=1fi(Tbibx)).(JΠj=1gj(Snjyj)), where (b1, b2, ..., bH)∊ZH and J is a positive integer. The functions fi and gj are bounded and the systems (X, F, μ, T) and (Yj, Gj, mj, Sj) are weakly mixing

Rota's alternating procedure with non-positive operators

AbstractLet (Tn) be a sequence of linear contractions on all Lp spaces, 1 ⩽ p ⩽ ∞. We show that limn T1∗T2∗ … Tn∗Tn … T2T1f exists a.e. for each function fϵL Log L. This extends to the non-positive case (G. C. Rota, Bull. Amer. Math. Soc.68, 95–102; N. Starr, Trans. Amer. Math. Soc.121, 90–115). We obtain also the a.e. convergence of products JvT1∗ … Tn∗JμTn … T1f in Lp for some non-positive contractions on Lp, 1 <p < ∞

A weighted pointwise ergodic theorem

We prove the following weighted ergodic theorem: Let (Xn) be an i.i.d. sequence of symmetric random variables such that E(|X1|p) < ∞ for some p, 1 < p < ∞. Then there exists a set of full measure Ω such that for ω ∈ Ω the following holds: For all dynamical systems(Y,g,ν,S), for all r, 1 < r ≤ ∞ and g ∈ Lr(ν) the averages 1/NΣn=1NXn(ω)g(Sny) converge p.s. ν

Pointwise characteristic factors for the multiterm return times theorem

This paper is an update and extension of a result the authors first proved in 2003. The goal of this paper is to study factors which are known to be L^2-characteristic for certain nonconventional averages and prove that these factors are pointwise characteristic for the multidimensional return times averages.Comment: 31 pages Submitted and Accepted to ETDS Dan Rudolph Memorial Volume. This version is the final accepted copy and includes all of the revisions recommended by the referee

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v