PCMEA: Pointwise Convergence of Multiple Ergodic Averages

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Szemerédi定理的遍历理论证明。关于多重遍历平均在非线性多项式轨道上的第一次突破是由H.Furstenberg和B.Weiss于20世纪90年代提出的，这导致了一系列的研究，最终导致M.Walsh著名的收敛结果，即多重遍历平均范数收敛的最优收敛结果。事实上，单函数沿多项式轨道的逐点收敛问题一直悬而未决，直到Bourain在20世纪80年代末和90年代初的突破性工作中；范数收敛的问题源于初等的Hilbert空间正交性论证。在最近与M.Mirek(罗格斯大学)和T.Tao(UCLA)的工作中，我建立了多重(非线性)多项式遍历平均范畴的第一个逐点收敛结果，即逐点双线性多项式遍历平均定理。这个建议的主题是发展多项式遍历平均的逐点收敛理论，尽可能类似于Walsh的结果。虽然提出的问题是动态的，但这一研究的结果将导致加法组合学、组合数论和调和分析的重大进步。

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.