Pointwise Convergence of Multiple Ergodic Averages

遍历理论是对非常普遍的等分布现象的研究：给出一个有界环境X的典型的‘随机化’但大小保持不变的变换T，人们试图理解轨道O(P)的性质：对于典型的初始位置p={p，Tp，T^2p，...}。为了使讨论严格，人们将测量理论结构强加给X，为它配备一个概率测量，即抽象的体积元素m。从我们在数值积分方面的工作，我们期望如果轨道O(P)关于m是‘装备分布的’，那么采样过程F_N(P)：=1/N(f(Tp)+f(T^2p)+…+f(T^Np))应该逼近f关于m的积分。1931年证明的George Birkhoff逐点遍历定理的内容是，如果f的积分存在，且T充分随机化，F_N(P)总是趋于f关于m的积分。通俗地说：f的“时间平均”收敛到F的“空间平均”。遍历理论中的一个经典问题涉及Birkhoff定理的多项式推广：G_N(P)的收敛：=1/N(g(T^{P(1)}p)+g(T^{P(2)}p)+...+g(T^{P(N)}p))，其中P是整系数多项式：即当限制到多项式次数时O(P)的等分布。在20世纪80年代末和90年代初，菲尔兹奖牌获得者Jean Bourain证明了：如果(比方说)当g有界时，除了m-可忽略的起始位置集外，G_N(P)也收敛；为了恢复g的积分，我们需要从我们的变换中稍微多一些随机化的行为。这个建议将研究通过研究许多不同函数、{h_1，…，h_m}许多不同的交换变换，{T_1，...，T_m}和许多不同次数的具有整系数的不同次数的多项式之间的干扰而形成的多个遍历平均的收敛性质。P_m}：我们将试图理解平均值K_N(P)的收敛：=1/N(H_1(P)+H_2(P)+…+H_N(P))其中h_n(P)：=h_1(T_1^{P_1(N)}p)x…Xh_m(T_m^{P_m(N)}p)。

Ergodic theory is the study of equidistribution phenomena in very great generality: given a typically ``randomizing," but size preserving transformation, T, of a bounded environment, X, one seeks to understand the properties of the orbit O(p) := {p, Tp, T^2p, ... } for a typical initial position, p. To make this discussion rigorous, one imposes measure theoretic structure on X, equipping it with a probability measure, i.e. an abstract volume element, m.From our work on numerical integration, we expect that if the orbit O(p) is ``equidistributed" with respect to m, then the sampling procedureF_N(p) := 1/N (f(Tp) + f(T^2 p) + ... + f(T^N p) )should approximate the integral of f with respect to m. The content of George Birkhoff's pointwise ergodic theorem, proven in 1931, is that aside from an m-negligible set of starting locations, F_N(p) always tends towards the integral of f with respect to m, provided that the integral of f exists, and T is sufficiently randomizing. Colloquially: the "time averages" of f converge to the "space average" of f.A classical problem in ergodic theory concerns polynomial extension of Birkhoff's theorem: the convergence of G_N(p) := 1/N ( g(T^{P(1)} p) + g(T^{P(2)} p) + ... + g(T^{P(N)} p) ), where P is a polynomial with integer coefficients: i.e. the equidistribution of O(p) when restricted to polynomial times. In the late 1980s and early 1990s, the Fields Medalist Jean Bourgain proved that provided that (say) whenever g is bounded, aside from an m-negligible set of starting locations, G_N(p) also converge; in order to recover the integral of g, we require slightly more randomizing behavior from our transformation, T.This proposal will study the convergence properties of multiple ergodic averages, formed by studying interference between many different functions, {h_1,...,h_m} many different commuting transformations, {T_1,...,T_m} and many different polynomials with distinct degrees with integer coefficients, {P_1,...,P_m}: we will seek to understand the convergence of the averagesK_N(p) := 1/N( H_1(p) + H_2(p) + ... + H_N(p))whereH_n(p) := h_1(T_1^{P_1(n)} p) x ... x h_m(T_m^{P_m(n)} p).