PCMEA: Pointwise Convergence of Multiple Ergodic Averages

对多重遍历平均的研究有着悠久而丰富的历史，至少可以追溯到H. Furstenberg对szemersamedi定理的遍历理论证明。20世纪90年代，H. Furstenberg和B. Weiss首次突破了沿非线性多项式轨道的多重遍历平均，引发了一系列研究，最终得出了M. Walsh著名的收敛结果，这是多重遍历平均范数收敛范畴中的最优收敛结果。理解逐点设置中的收敛现象要微妙得多。事实上，单函数沿多项式轨道的点向收敛问题一直是开放的，直到20世纪80年代末和90年代初布尔甘的突破性工作；范数收敛问题由初等希尔伯特空间正交性论证而来。在最近与M. Mirek（罗格斯大学）和T. Tao（加州大学洛杉矶分校）的合作中，我建立了多个（非线性）多项式遍历平均类别中的第一个逐点收敛结果，即逐点双线性多项式遍历定理。本提案的主题是发展多多项式遍历平均的点向收敛理论，使其尽可能接近于沃尔什的结果。虽然所提出的问题是动态的，但这条研究路线的结果将导致加性组合学、组合数论和调和分析的重大进展。

The study of multiple ergodic averages has a long and rich history, dating back at least to H. Furstenberg's ergodic theoretic proof of Szemerédi's Theorem. The first breakthrough in multiple ergodic averages along non-linear polynomial orbits was due to H. Furstenberg and B. Weiss in the 1990s, which lead to a flurry of research, ultimately leading to M. Walsh's celebrated convergence result, an optimal convergence result in the category of norm convergence of multiple ergodic averages.Understanding convergence phenomena in the pointwise setting is far more subtle. Indeed, the issue of pointwise convergence of single function along a polynomial orbit remained open until breakthrough work of Bourgain in the late 1980s and early 1990s; the issue of norm convergence follows from elementary Hilbert space orthogonality arguments.In recent work with M. Mirek (Rutgers University) and T. Tao (UCLA), I established the first pointwise convergence result in the category of multiple (non-linear) polynomial ergodic averages, namely a pointwise bilinear polynomial ergodic theorem.The theme of this proposal is to develop the theory of pointwise convergence of multiple polynomial ergodic averages in as close an analogy to Walsh's result as possible. Although the questions posed are dynamical, results of this line of inquiry will lead to major advances in additive combinatorics, combinatorial number theory, and harmonic analysis.