Convergence of Conze–Lesigne averages

By B. Host, Bryna Kra. Ergodic Theory and Dynamical Systems, 2001

50 citations recorded by the source corpus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7a65abcb2babfe412ed09a5f7ff7339dfcce1936

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Citations in the corpus

The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view (2005, unknown): “It seems that finding the exact link between the anti-uniform functions of level k and the k-step nilsystems found in the work of Host and Kra [21] would clarify the connections between the two fields and probably lead to new and interesting developments.”
Universal characteristic factors and Furstenberg averages (2004, methodology): “The factor corresponding to arithmetic progressions of length 4, an inverse limit of 2-step nilﬂows, was studied by Conze–Lesigne [CL84], [CL87], [CL88], Furstenberg–Weiss [FuW96], and Host–Kra [HK01], [HK02], and hinted to the nature of the complete series.”
The primes contain arbitrarily long arithmetic progressions (2004, methodology): “sition 2.3 is Proposition 3.5 below. Once Proposition 3.5 is proved, we turn to the issue of ﬁnding primes in AP. A possible choice for νwould be Λ, the von Mangoldt function (this is deﬁned to equal [24]-[26], [39]), might be analogous to a kind of “higher order Fourier analysis” which could be use”
An inverse theorem for the Gowers U^3 norm (2005, background): “ll be our key reference. However as we shall see later there are some strong connections between this approach and the ergodic one, especially after the work on characteristic factors by Host and Kra [39, 41] and Ziegler [61, 62], and on the connection to nilsequences by Bergelson, Host, and Kra [3]”
The dichotomy between structure and randomness, arithmetic progressions, and the primes (2005, background): “More recently, a more efficient " hard " factor Z 2 was constructed by Conze– Lesigne [7], Furstenberg–Weiss [14], and Host–Kra [38]; the analogous factors for Terence Tao higher k are more difficult to construct, but this was achieved by Host–Kra in [39], and also subsequently by Ziegler [64].”
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.”
Some open problems on multiple ergodic averages (2011, background): “Mean convergence of the averages (1) was established after a long series of intermediate results; the papers [26, 16, 17, 18, 28, 41, 30, 31, 45] dealt with the important case of linear polynomials, and using the machinery introduced in [31], convergence for arbitrary polynomials was finally obtaine”
On the norm convergence of non-conventional ergodic averages (2008, methodology): “These, in turn, build on techniques developed in previous pa pers for this or other special cases of the theorem by Conze and Lesigne [4, 5, 6], Zh ang [18] and Host and Kra [12], and also on the analysis by Furstenberg and Weis s in [9] of averages 2”