Pointwise Convergence of Multiple Ergodic Averages

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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).