Nonconventional ergodic averages and nilmanifolds

By B. Host, Bryna Kra. 2005

494 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/2fbf1c7504be56270f6300d69fe83dadcaf350d3

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In Algebra and Number Theory

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Citations in the corpus

A non-conventional ergodic theorem for a nilsystem (2002, background): “It was shown by Host and Kra [6], that k-step nilsystems ‘characterize’ the L2 behavior of the averages in (1) (this was shown for k = 3 by Furstenberg and Weiss [4] and Conze and Lesigne [2], and for k = 4 by Host and Kra [5] and by the present author in her PhD thesis [17]).”
Higher order Fourier analysis of multiplicative functions and applications (2014, result): “Similar comments apply for analogous decomposition results in ergodic theory [30].”
Multiple recurrence for two commuting transformations (2009, background): “n1 1 f·T n2 2 f·T n1 1 T n2 2 fdµ 1/4 . In fact, we can take the simultaneous instead of iterated limit by Theorem 1.1 of [5]. By the H¨older Inequality, Lemma 3.2 implies that, for every f ∈ L∞(µ), (13) |||f|||T1,T2 ≤ kfkL4(µ) . Let π: (X,µ,T1,T2) → (Y,ν,S1,S2) be a factor map, and f∈ L∞(ν), by Lem”
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle (2008, background): “These results can be viewed as a finite field analogue of the results in [14] in high characteristic (and a partial analogue in the low characteristic case), and indeed draws heavily on the tools developed in that paper; see [2] for further discussion.”
Pointwise convergence of ergodic averages along cubes (2010, background): “We refer to [11] for the definition of these nilsystems (and for a nilsystem in general).”
The quantitative behaviour of polynomial orbits on nilmanifolds (2007, methodology): “In a companion paper [13] we shall use this theorem to establish the Möbius and Nilsequences conjecture from our earlier paper [12].”
The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view (2005, methodology): “These norms are used by Gowers (as well as by Host and Kra [21] and more recently by Green and Tao [16] and by Tao [28]) to control the average along arithmetic progressions, meaning the quantity in Equation (4.”
Convergence of multiple ergodic averages for some commuting transformations (2004, methodology): “ergodic. Then S ∈ G(X,T), G(X,T) = G(X,S), and for all integers k ≥ 1 and all f ∈ L∞(µ), µ[k] T = µ [k] S , I [k](T) = I[k](S), |||f||| k,T = |||f|||k,S, and Zk(X,T) = Zk(X,S). Proof. By Lemma 5.5 in [HK03], S ∈ G(X,T). We use induction on k to show that µ[k] T = µ [k] S and I [k](T) = I[k](S). The ”
Norm convergence of nilpotent ergodic averages (2011, background): “The study of this case originated in the seminal work of Furstenberg [10] on Szemerédi’s theorem, while a general solution when the polynomials are linear was later provided by Host and Kra [14] following the work of several authors (with a different proof subsequently found by Ziegler [22]).”
The dichotomy between structure and randomness, arithmetic progressions, and the primes (2005, background): “In principle one could use higher order Fourier analysis (or the precise characteristic factors achieved in [39], [64] to obtain this decomposition, but this looks rather difficult technically, though progress has been made in the k = 4 case.”
Bilinear Hilbert transforms along curves, I: the monomial case (2008, methodology): “For simplicity, we assume ρ is supported on [1/8, 2].”
On exchangeable random variables and the statistics of large graphs and hypergraphs (2008, background): “In addition, Razborov is by no means the first to consider a notion of exchangeable probability measure on a space of models of a simple theory; for example, these appear already in the studies by Gaifman [34] and Krauss [51], and their work is then offered as motivation in Hoover’s article [42].”