Convergence of multiple ergodic averages along polynomials of several variables

By Alexander Leibman. 2005

121 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/6b3ae9e2803cc10881c608e113b22465ea2b4231

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Ergodic seminorms for commuting transformations and applications (2008, methodology): “The same structure can be used to study other problems of multiple convergence and of multiple recurrence, for example in [HK2], [ L ], [BHK], [FHK], [FK2], ... A similar method was used by Conze & Lesigne for two commuting transformations, but all attempts to solve the general case by using the mac”
Multiple ergodic averages for three polynomials and applications (2006, unknown): “Using methods from [19], this was done in [20] for weak convergence, and in [25] for strong convergence of the averages ( P ).”
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]). If q(n) is an integer valued polynomial, the existence of the limit for {q(n): n ∈ N} is shown in [16] (see [17] for a more general result). It was a folklore theorem that for k = 1, the limit exists for the ”
Pointwise ergodic theorems for non-conventional bilinear polynomial averages (2020, background): “p 1 + + 1 p k 1. Then the averages AP 1;:::;P k N (f 1;:::;f k) converge in Lp(X) norm for any 0 <p<1with 1 p 1 + + 1 p k <1 p . Proof. The case p 1 = = p k= 1, p= 2 is established in [39], [50] (see also [73] for a quite dierent proof and a generalization); one can then extend to other 0 &”
Intersective polynomials and polynomial Szemeredi theorem (2007, background): “It was proved in [L4] (see also [HK2]) that for any probability measure preserving system ( X, T, µ ), a certain pro-nilsystem ( X, T , µ ) is a characteristic factor of ( X, T, µ ) with respect to the system of polynomial actions { T p 1 ( n ) , . . . e e , T p r ( n ) } , which means that ( X, T…”
The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view (2005, background): “, fk are known to exist in L 2(μ) (see [22] and [23]).”
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].”
Norm convergence of nilpotent ergodic averages (2011, background): “We will make use of the following results of Leibman.”
Some open problems on multiple ergodic averages (2011, background): “Leibman [35] that gives useful information about their structure.”