Extension of Wiener-Wintner double recurrence theorem to polynomials

Abstract

We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case where we have a polynomial exponent. We will show that there exists a single set of full measure for which the averages 1 N N

Weighted multiple ergodic averages and correlation sequences

“…Since G (i 1 ,...,is) is a subgroup of the torus G s and p 1 , p ′ 1 , p 2 , p ′ 2 are distinct primes, it follows that the group G (i 1 ,...,is) is finite. Furthermore, since G = G (1) G (2) , using Lemma 8.3 and induction, we have that the group G s is the product of the groups G (i 1 ,...,is) for i 1 , . .…”

“…(ii) Bounded generalized polynomials (see [48,Corollary 2.23]). 2 These include sequences of the form ({p(n)}) n∈N or (e(p(n))) n∈N , where p : N → Z is an arbitrary generalized polynomial, {x} denotes the fractional part of x, and e(t) := e 2πit .…”

“…All the groups [H (i) , Q (j) ] are normal and included in [H, Q], thus it suffices to prove that [H, Q] is contained in the product of these groups. If h i ∈ H (i) for i = 1, 2 and q ∈ Q (1) , it follows from (8.4) and from the normality of [H (1) , Q (1) ] that [h 1 h 2 , q] ∈ [H (1) , Q (1) ] · [H (2) , Q (1) ]. This proves the result in the case k = 2, ℓ = 1.…”

“…When d = 1, examples of good universal weights for some multiple ergodic averages can be found in [1,2,3,4,19,26,30,32,48,70]. Most of these results deal with the case where ℓ = 1 and are based on the theory of characteristic factors that was pioneered by H. Furstenberg.…”

On the homogeneous ergodic bilinear averages with Möbius and Liouville weights

“…Therein, the authors proved a Wiener-Wintner version of BDRT, that is, the exponential sequences (e 2πint ) n∈Z are good weight for the homogeneous ergodic bilinear averages. Subsequently, I. Assani and R. Moore showed that the polynomials exponential sequences e 2πiP (n) n∈Z are also uniformly good weights for the homogeneous ergodic bilinear averages [8]. One year later, I. Assani [9] and P. Zorin-Kranich [28] proved independently that the nilsequences are uniformly good weights for the homogeneous ergodic bilinear averages.…”