Universal characteristic factors and Furstenberg averages

By T. Ziegler. 2004

264 citations recorded by the source corpus.

Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and a_j are integers. A factor of X is characteristic for averaging schemes of length k (or k-characteristic) if for any non zero distinct integers a_1,...,a_k, the limiting L^2(\mu) behavior of the averages in (*) is unaltered if we first project the functions f_j onto the factor. A factor of X is a k-universal characteristic factor (k-u.c.f)} if it is a k-characteristic factor, and a factor of any k-characteristic factor. We show that there exists a unique k-u.c.f, and it has a structure of a (k-1)-step nilsystem, more specifically an inverse limit of (k-1)-step nilflows. Using this we show that the averages in (*) converge in L^2(\mu). This provides an alternative proof to the one given by Host and Kra in 2002.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/307b7c4232a333633dccaef74f8f7a2097626803

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Citations in the corpus

The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view (2005, background): “More recently, Ziegler [32] has an alternate approach for all k.”
Pointwise ergodic theorems for non-conventional bilinear polynomial averages (2020, unknown): “, [38], [75], [2], [5], [27], [32]) towards Theorem 1.”
Infinite-step nilsystems, independence and complexity (2011, background): “are d-step nilsystems for some integer d ≥ 1 (see also [34]).”
Maximal multilinear operators (2005, background): “The question of norm convergence of T A,X,L is more difficult and has only been recently treated in the independent works of Host and Kra [12] and Ziegler [21].”
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]).”
An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement) (2010, background): “See also [22, 36] for related work on multiple averages and nilmanifolds in ergodic theory.”
Multiple recurrence and convergence for sequences related to the prime numbers (2006, unknown): “In [15], it is shown that the limit exists for sn = n (another proof is given in [23]).”
The primes contain arbitrarily long arithmetic progressions (2004, methodology): “se). 12The situation again has an intriguing parallel with ergodic theory, in which the rˆole of the antiuniform functions of order k − 2 appear to be played by k − 2-step nilfactors (see [25], [26], [39]), which may contain polynomial eigenfunctions of order k−2, but can also exhibit slightly more ”
Multiple ergodic averages for three polynomials and applications (2006, unknown): “It was shown in [35] that Z k − 1 is in fact the smallest characteristic factor for the family { l 1 n, l 2 n, . . . , l k n } .”
Multiple recurrence and nilsequences (2005, background): “The lim inf is actually a limit; see [HK]. (See also [ Z2 ].)”
The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic (2011, unknown): “To classify polynomials of higher degree we employ the method of vertical differentiation, which is used extensively in the ergodic theory literature (see for instance [26], [46], [3]); for this simple example we use a very concrete instance of this method here.”
Norm convergence of multiple ergodic averages for commuting transformations (2007, methodology): “In particular, the l = 2 case of our arguments are a finitary analogue of those in [3].”
Some open problems on multiple ergodic averages (2011, background): “, `n}) = ` − 1 ([31, 45]), and dmin(P) = 1 when P consists of at least two rationally independent polynomials ([22, 24]).”
Multiple recurrence and convergence for hardy sequences of polynomial growth (2009, background): “Nevertheless, we now have several different proofs of convergence in the case where the sequences are linear ([26], [38], [33], [35],[1], [25]), and the case where all the sequences are integer polynomials was treated in [27] and [29].”
Nil Bohr-sets and almost automorphy of higher order (2014, background): “This question was finally answered by Host and Kra in [34] (see also Ziegler in [56]).”
Growth in groups: ideas and perspectives (2013, unknown): “eams, one coursing through work on arithmetic progressions by Schur, van der Waerden, Roth [Rot53], Szemer´edi [Sze69], Furstenberg [Fur77] (leading to the ergodic work of Host-Kra [HK05] and Ziegler [Zie07]; see also Szegedy [Sze]), Gowers [Gow01], and Green-Tao [GT08], among many others, and anoth”