Nikos Frantzikinakis
In 1993, E. Lesigne proved a polynomial extension of the Wiener-Wintner ergodic theorem and asked two questions: does this result have a uniform counterpart and can an assumption of total ergodicity be replaced by ergodicity? The purpose of this article is to answer these questions, the first one positively and the second one negatively.
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We extend the authors' previous work on Wiener-Wintner double recurrence theorem to the case of polynomials.
We show uniform convergence of Wiener-Wintner ergodic averages for ergodic actions of (not necessarily countable) locally compact, second countable, abelian (LCA) groups. As a by-product, we obtain a finitary version of the van der Corput inequality for such groups.
For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry. The result applies to ergodic nilsystems. Fully oscillating sequences can then be constructed on nilmanifolds.
We give a Fourier analytic proof of the generalisation due to Host and Kra of the classical Wiener-Wintner theorem and give some explicit bounds on the limit of the weighted ergodic averages.
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.
More stability and convergence results for higher-order Wiener-Wintner systems
该研究针对幂型高阶Wiener-Wintner系统，证明高阶Wiener-Wintner平均的次线性性及对条件期望、乘积的界定，推导其在和、因子、乘积下的更优稳定性，还得到多项式返回时间定理等更一般的收敛结果。
We show that a $k$-linear pointwise ergodic theorem on an ergodic measure-preserving system implies a uniform $k$-linear nilsequence Wiener-Wintner theorem on that system. The assumption is known to hold for arbitrary systems and $k=2$ (due to Bourgain) and for distal systems and arbitrary $k$ (due to Huang, Shao, and Ye).
In this paper, we extend Bourgain's double recurrence result to the Wiener-Wintner averages. Let $(X, \mathcal{F}, μ, T)$ be a standard ergodic system. We will show that for any $f_1, f_2 \in L^\infty(X)$, the double recurrence Wiener-Wintner average \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x) e^{2πi n t} \] converges off a single null set of $X$ independent of $t$ as $N \to \infty$. Furthermore, we will show a uniform Wiener-Wintner double recurrence result: If either $f_1$ or $f_2$ belongs to the orthogonal complement of the Conze-Lesigne factor, then there exists a set of full meas…
Wiener--Wintner points for topological dynamical systems
该文研究局部紧阿贝尔群上的动力系统，将Wiener-Wintner型平均收敛性与系统特征值关联，提出Wiener--Wintner点概念并证明其遍历情形下满测度，还明确了其在非周期序模型中与衍射理论的对应关系。
We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the $\mathbb{R}$-action which assert that for any family of maps $(T_t)_{t \in \mathbb{R}}$ acting on the Lebesgue measure space $(Ω,{\cal {A}},μ)$ where $μ$ is a probability measure and for any $t\in \mathbb{R}$, $T_t$ is measure-preserving transformation on measure space $(Ω,{\cal {A}},μ)$ with $T_t \circ T_s =T_{t+s}$, for any $t,s\in \mathbb{R}$. Then, for any $f \in L^1(μ)$, there is a a single null set off which $\displaystyle \lim_{T \rightarro…
For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such systems are abundant in a precise sense. We show that the CF-Nil($k$) systems are precisely the class of minimal systems for which the $k$-step nilsequence version of the Wiener-Wintner average converges everywhere. As part of the proof we establish that pronilsystems are $coalescent$ both in the measurable and topological categories. In addition, we character…
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 \[ \frac{1}{N} \sum_{n=1}^N f_1(T^{an}x)f_2(T^{bn}x)ϕ(p(n)) \] converge for any polynomial $p$ with real coefficients, and any continuous function $ϕ$ from the torus to the set of complex numbers . We also show that if either function belongs to an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends on the degree of the polynomial $p$, then the averages co…
Higher Order Wiener-Wintner systems: examples and applications
本研究构建高维Wiener-Wintner动力系统，通过扩展Bourgain的双重回归估计实现多重遍历平均的定量控制，给出多重回归相关平均几乎处处收敛的简洁证明，并提供Host-Kra-Ziegler因子的新特征。
Inspired by topological Wiener-Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on $C(K)$ and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.
In this article, we obtain a version of the noncommutative Banach Principle suitable to prove Wiener-Wintner type results for weights in W1-space. This is used to obtain noncommutative Wiener-Wintner type ergodic theorems for various types of weights for certain types of positive Dunford-Schwartz operators. We also study the b.a.u. (a.u.) convergence of some subsequential averages and moving averages of such operators
Quantitative Polynomial Wiener-Wintner Theorems
在包括幂零李群保测作用的广泛框架下，证明了定量多项式Wiener-Wintner定理，适用于遍历平均和奇异积分权重平均，证明依赖于广义多项式Carleson定理。
We prove the following extension of the Wiener--Wintner Theorem in Ergodic Theor and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows $ (X,μ, T_t)$ and $ f\in L^p (X,μ)$, there is a set $X_f\subset X $ of probability one, so that for all $x\in X_f$ we have \begin{equation*} \lim _{s\downarrow0} \int _{s<\abs t<1/s} \operatorname e ^{i θt} f(\operatorname T_tx)\; \frac{dt}t \qquad \text{exists for all $θ$.} \end{equation*} The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson's theorem.
Multilinear Wiener-Wintner type ergodic averages and its application
本文将 Host-Kra 广义 Wiener-Wintner 定理推广至多重线性情形，在逐点收敛假设下证明了对任意保测系统、不同非零整数及零序列，相应多重线性平均极限存在。
An extension of the Wiener-Wintner ergodic theorem for pointwise jointly ergodic systems and its applications
该研究将经典Wiener-Wintner遍历定理推广到带nilsequence权的逐点联合遍历系统，并给出其在加权遍历平均收敛、沿⌊αn⌋子序列遍历平均几乎处处收敛方面的应用。
A Hard-Analytic Proof of "Most" Polynomial Wiener-Wintner Theorems for Infinite Measure Spaces
本文使用硬分析方法，为σ-有限测度空间下“多数”多项式Wiener-Wintner定理提供了新证明，指出在保测系统中特定多项式加权和的收敛性。
In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener-Wintner type ergodic theorem for dynamical systems not necessarily being commutative. More precisely, we introduce a weight class $\mathcal{D}$, which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set $$Λ_d=\left\{\{λ^{k_1}_1\dotsmλ^{k_d}_d\}_{(k_1,\dots,k_d)\in \mathbb{N}^d}:\quad (λ_1,\dots,λ_d) \in \mathbb{T}^d\right\};$$ then prove a multi-parameter Bellow and Losert's Wiener-Wintner type ergodic theorem for the class $\mathcal{D}$ and for noncommutati…
Non-Commutative Wiener-Wintner theorem for amenable group actions
本文针对有限von Neumann代数上保迹自同构的可数群作用，建立了Jacobs-de Leeuw-Glicksberg分解，并证明了非交换Wiener-Wintner定理。
The Wiener Wintner Theorem Along the Primes
证明沿素数序列的维纳 - 温特纳定理，这是该定理首次扩展到算术序列。通过经典傅里叶分析等多理论结合，利用\(U^3\)理论，得出对给定概率空间及函数\(f\)，存在全概率集使特定和式对所有\(\theta\in[0,1]\)收敛。
A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on $\Z/N\Z$ introduced by Gowers in his proof of Szemerédi's Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg's proof of Szemerédi's Theorem) defined by the authors. For each integer $k\geq 1$, we define seminorms on $\ell^\infty(\Z)$ analogous to these norms and seminorms. We study t…
Inspired by subsequential ergodic theorems, we study the validity of Wiener's lemma and the extremal behavior of a measure $μ$ on the unit circle via the behavior of its Fourier coefficients $\hatμ(k_n)$ along subsequences $(k_n)$. We focus on arithmetic subsequences such as polynomials, primes and polynomials of primes, and also discuss connections to rigidity sequences, return times sequences and strongly sweeping out sequences as well as measures on $\mathbb{R}$. We also present consequences for orbits of operators and of $C_0$-semigroups on Hilbert and Banach spaces extending the results o…
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