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Bryna Kra - One of the best experts on this subject based on the ideXlab platform.
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Convergence of polynomial Ergodic averages
Israel Journal of Mathematics, 2005Co-Authors: Bernard Host, Bryna KraAbstract:We prove the L ^2 convergence for an Ergodic average of a product of functions evaluated along polynomial times in a totally Ergodic System. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilSystems, that controls the limit behavior of the average. For a general System, we prove the convergence for certain families of polynomials.
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Multiple recurrence and nilsequences
Inventiones mathematicae, 2005Co-Authors: Vitaly Bergelson, Bernard Host, Bryna Kra, Imre RuzsaAbstract:Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving System $(X,\mathcal{X},\mu,T)$ and a set $A\in\mathcal{X}$ of positive measure, the set of integers n such that $\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length ( k +1) of the arithmetic progression under consideration. In an Ergodic System, for k =2 and k =3, this set is syndetic, while for k ≥4 it is not. The main tool is a decomposition result for the multicorrelation sequence $\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}$ , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d ^*( E )>0 and for all ε>0, the set $$\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}$$ is syndetic.
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Multiple recurrence and nilsequences
Inventiones Mathematicae, 2005Co-Authors: Vitaly Bergelson, Bernard Host, Bryna KraAbstract:Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving System $(X,\CX,\mu,T)$ and a set $A\in\CX$ of positive measure, the set of integers $n$ such that $\mu(A\cap T^nA\cap T^{2n}A\cap \ldots\cap T^{kn}A) >\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length $(k+1)$ of the arithmetic progression under consideration. In an Ergodic System, for $k=2$ and $k=3$, this set is syndetic, while for $k\geq 4$ it is not. The main tool is a decomposition result for the multicorrelation sequence $\int f(x)f(T^nx)f(T^{2n}x)\ldots f(T^{kn}x) \,d\mu(x)$, where $k$ and $n$ are positive integers and $f$ is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers $E$ with upper Banach density $d^*(E)>0$ and for all $\epsilon > 0$, the set $$ \{n\in\Z\colon d^*\bigl(E\cap (E+n)\cap (E+2n)\cap (E+3n)\bigr)> d^*(E)^4-\epsilon\}$$ is syndetic.
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Polynomial averages converge to the product of integrals
Israel Journal of Mathematics, 2005Co-Authors: Nikos Frantzikinakis, Bryna KraAbstract:We answer a question posed by Vitaly Bergelson, showing that in a totally Ergodic System, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL2 to the product of the integrals. Such averages are characterized by nilSystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds.
Bernard Host - One of the best experts on this subject based on the ideXlab platform.
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Convergence of polynomial Ergodic averages
Israel Journal of Mathematics, 2005Co-Authors: Bernard Host, Bryna KraAbstract:We prove the L ^2 convergence for an Ergodic average of a product of functions evaluated along polynomial times in a totally Ergodic System. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilSystems, that controls the limit behavior of the average. For a general System, we prove the convergence for certain families of polynomials.
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Multiple recurrence and nilsequences
Inventiones mathematicae, 2005Co-Authors: Vitaly Bergelson, Bernard Host, Bryna Kra, Imre RuzsaAbstract:Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving System $(X,\mathcal{X},\mu,T)$ and a set $A\in\mathcal{X}$ of positive measure, the set of integers n such that $\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length ( k +1) of the arithmetic progression under consideration. In an Ergodic System, for k =2 and k =3, this set is syndetic, while for k ≥4 it is not. The main tool is a decomposition result for the multicorrelation sequence $\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}$ , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d ^*( E )>0 and for all ε>0, the set $$\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}$$ is syndetic.
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Multiple recurrence and nilsequences
Inventiones Mathematicae, 2005Co-Authors: Vitaly Bergelson, Bernard Host, Bryna KraAbstract:Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving System $(X,\CX,\mu,T)$ and a set $A\in\CX$ of positive measure, the set of integers $n$ such that $\mu(A\cap T^nA\cap T^{2n}A\cap \ldots\cap T^{kn}A) >\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length $(k+1)$ of the arithmetic progression under consideration. In an Ergodic System, for $k=2$ and $k=3$, this set is syndetic, while for $k\geq 4$ it is not. The main tool is a decomposition result for the multicorrelation sequence $\int f(x)f(T^nx)f(T^{2n}x)\ldots f(T^{kn}x) \,d\mu(x)$, where $k$ and $n$ are positive integers and $f$ is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers $E$ with upper Banach density $d^*(E)>0$ and for all $\epsilon > 0$, the set $$ \{n\in\Z\colon d^*\bigl(E\cap (E+n)\cap (E+2n)\cap (E+3n)\bigr)> d^*(E)^4-\epsilon\}$$ is syndetic.
Wen Huang - One of the best experts on this subject based on the ideXlab platform.
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Pointwise convergence of multiple Ergodic averages and strictly Ergodic models
Journal D Analyse Mathematique, 2019Co-Authors: Wen Huang, Song Shao, Xiangdong YeAbstract:By building some suitable strictly Ergodic models, we prove that for an Ergodic System $$(X, \mathcal{X}, \mu, T), d \in \mathbb{N}, f_1, \ldots, f_d \in L^\infty(\mu)$$, the averages $$\frac{1}{{{N^2}}}\sum\limits_{(n,m) \in {{[0,N - 1]}^2}} {{f_1}({T^n}x){f_2}({T^{n + m}}x) \cdots {f_d}({T^{n + (d - 1)m}}x)}$$ converge to a constant μ a.e. Deriving some results from the construction, for distal Systems we answer positively the question if the multiple Ergodic averages converge a.e. That is, we show that if $$(X, \mathcal{X}, \mu, T)$$ is an Ergodic distal System, and f1, …, fd ∈ L∞(μ), then the multiple Ergodic averages $$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{f_1}({T^n}x) \cdots {f_d}({T^{dn}}x)}$$ converge μ a.e.
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Möbius disjointness for topological models of Ergodic Systems with discrete spectrum
Journal of Modern Dynamics, 2019Co-Authors: Wen Huang, Zhiren Wang, Guohua ZhangAbstract:We provide a criterion for a point satisfying the required disjointness condition in Sarnak's Mobius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an Ergodic System with discrete spectrum.
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Strictly Ergodic Models Under Face and Parallelepiped Group Actions
Communications in Mathematics and Statistics, 2017Co-Authors: Wen Huang, Song ShaoAbstract:The Jewett–Krieger theorem states that each Ergodic System has a strictly Ergodic topological model. In this article, we show that for an Ergodic System one may require more properties on its strictly Ergodic model. For example, the orbit closure of points in diagonal under face transforms may be also strictly Ergodic. As an application, we show the pointwise convergence of Ergodic averages along cubes, which was firstly proved by Assani (J Anal Math 110:241–269, 2010).
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M\"{o}bius disjointness for topological models of Ergodic Systems with discrete spectrum
arXiv: Dynamical Systems, 2016Co-Authors: Wen Huang, Zhiren Wang, Guohua ZhangAbstract:We provide a criterion for a point satisfying the required disjointness condition in Sarnak's Mobius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an Ergodic System with discrete spectrum.
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Pointwise convergence of multiple Ergodic averages and strictly Ergodic models
arXiv: Dynamical Systems, 2014Co-Authors: Wen Huang, Song Shao, Xiangdong YeAbstract:By building some suitable strictly Ergodic models, we prove that for an Ergodic System $(X,\mathcal{X},\mu, T)$, $d\in{\mathbb N}$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, the averages $$\frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2} f_1(T^nx)f_2(T^{n+m}x)\ldots f_d(T^{n+(d-1)m}x) $$ converge $\mu$ a.e. Deriving some results from the construction, for distal Systems we answer positively the question if the multiple Ergodic averages converge a.e. That is, we show that if $(X,\mathcal{X},\mu, T)$ is an Ergodic distal System, and $f_1, \ldots, f_d \in L^{\infty}(\mu)$, then multiple Ergodic averages $$\frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\ldots f_d(T^{dn}x) $$ converge $\mu$ a.e.
Vitaly Bergelson - One of the best experts on this subject based on the ideXlab platform.
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Multiple recurrence and nilsequences
Inventiones mathematicae, 2005Co-Authors: Vitaly Bergelson, Bernard Host, Bryna Kra, Imre RuzsaAbstract:Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving System $(X,\mathcal{X},\mu,T)$ and a set $A\in\mathcal{X}$ of positive measure, the set of integers n such that $\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length ( k +1) of the arithmetic progression under consideration. In an Ergodic System, for k =2 and k =3, this set is syndetic, while for k ≥4 it is not. The main tool is a decomposition result for the multicorrelation sequence $\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}$ , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d ^*( E )>0 and for all ε>0, the set $$\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}$$ is syndetic.
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Multiple recurrence and nilsequences
Inventiones Mathematicae, 2005Co-Authors: Vitaly Bergelson, Bernard Host, Bryna KraAbstract:Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving System $(X,\CX,\mu,T)$ and a set $A\in\CX$ of positive measure, the set of integers $n$ such that $\mu(A\cap T^nA\cap T^{2n}A\cap \ldots\cap T^{kn}A) >\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length $(k+1)$ of the arithmetic progression under consideration. In an Ergodic System, for $k=2$ and $k=3$, this set is syndetic, while for $k\geq 4$ it is not. The main tool is a decomposition result for the multicorrelation sequence $\int f(x)f(T^nx)f(T^{2n}x)\ldots f(T^{kn}x) \,d\mu(x)$, where $k$ and $n$ are positive integers and $f$ is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers $E$ with upper Banach density $d^*(E)>0$ and for all $\epsilon > 0$, the set $$ \{n\in\Z\colon d^*\bigl(E\cap (E+n)\cap (E+2n)\cap (E+3n)\bigr)> d^*(E)^4-\epsilon\}$$ is syndetic.
Imre Ruzsa - One of the best experts on this subject based on the ideXlab platform.
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Multiple recurrence and nilsequences
Inventiones mathematicae, 2005Co-Authors: Vitaly Bergelson, Bernard Host, Bryna Kra, Imre RuzsaAbstract:Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving System $(X,\mathcal{X},\mu,T)$ and a set $A\in\mathcal{X}$ of positive measure, the set of integers n such that $\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon$ is syndetic. The size of this set, surprisingly enough, depends on the length ( k +1) of the arithmetic progression under consideration. In an Ergodic System, for k =2 and k =3, this set is syndetic, while for k ≥4 it is not. The main tool is a decomposition result for the multicorrelation sequence $\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}$ , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d ^*( E )>0 and for all ε>0, the set $$\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}$$ is syndetic.