Multiple ergodic averages for three polynomials and applications

By Nikos Frantzikinakis. 2006

63 citations recorded by the source corpus.

We find the smallest characteristic factor and a limit formula for the multi- ple ergodic averages associated to any family of three polynomials and polynomial fam- ilies of the form {l1p,l2p,...,lkp}. We then derive several multiple recurrence results and combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemeredi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all " > 0 and every subset of the integersthe set

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

View Paper: https://www.semanticscholar.org/paper/eb4d735490702bded8d7fd27f564358742f13b0a

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In Discrete Mathematics and Combinatorics

1. Q. Chu
2. Nikos Frantzikinakis
3. B. Host
4. Terence Tao
5. T. Ziegler
6. Bryna Kra
7. M. N. Walsh
8. Tim Austin

In Geometry and Topology

1. Q. Chu
2. Nikos Frantzikinakis
3. B. Host
4. Terence Tao
5. T. Ziegler
6. Bryna Kra
7. Ben Joseph Green
8. Tim Austin

In Mathematical Physics

1. Q. Chu
2. Nikos Frantzikinakis
3. B. Host
4. Terence Tao
5. T. Ziegler
6. M. N. Walsh
7. Bryna Kra
8. Ben Joseph Green

Citations in the corpus

• Ergodic averages of commuting transformations with distinct degree polynomial iterates (2009, background): “er similar ergodicity assumptions, the result probably fails when 3 exponents agree no matter what exponent one uses on the right hand side (a conditional counterexample appears in Proposition 5.2 of [14]). It will be clear from our argument that in the statement of Theorem 1.3 we can replace the po”
• Some open problems on multiple ergodic averages (2011, background): “More examples of families P where dmin(P) has been computed can be found in [21, 36].”
• Multiple recurrence and convergence for hardy sequences of polynomial growth (2009, background): “If q ∈ Z[t] does not have zero constant term, then the sequence q(n) is good for multiple recurrence if and only if the range of the polynomial contains multiples of every positive integer ([13]).”
• Multiple recurrence for two commuting transformations (2009, methodology): “3 is a method introduced by Frantzikinakis [7].”