Sets of k -recurrence but not ( k + 1 ) -recurrence

Nikos Frantzikinakis[1]; Emmanuel Lesigne[2]; Máté Wierdl[3]

  • [1] Pennsylvania State University Department of Mathematics McAllister Building University Park, PA 16802 (USA)
  • [2] Université François Rabelais de Tours Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083) Faculté des Sciences et Techniques Parc de Grandmont 37200 Tours (France)
  • [3] University of Memphis Department of Mathematical Sciences Memphis, TN 38152 (USA)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 4, page 839-849
  • ISSN: 0373-0956

Abstract

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For every k ∈ ℕ , we produce a set of integers which is k -recurrent but not ( k + 1 ) -recurrent. This extends a result of Furstenberg who produced a 1 -recurrent set which is not 2 -recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.

How to cite

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Frantzikinakis, Nikos, Lesigne, Emmanuel, and Wierdl, Máté. "Sets of $k$-recurrence but not $(k+1)$-recurrence." Annales de l’institut Fourier 56.4 (2006): 839-849. <http://eudml.org/doc/10173>.

@article{Frantzikinakis2006,
abstract = {For every $k\in \mathbb\{N\}$, we produce a set of integers which is $k$-recurrent but not $(k+1)$-recurrent. This extends a result of Furstenberg who produced a $1$-recurrent set which is not $2$-recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.},
affiliation = {Pennsylvania State University Department of Mathematics McAllister Building University Park, PA 16802 (USA); Université François Rabelais de Tours Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083) Faculté des Sciences et Techniques Parc de Grandmont 37200 Tours (France); University of Memphis Department of Mathematical Sciences Memphis, TN 38152 (USA)},
author = {Frantzikinakis, Nikos, Lesigne, Emmanuel, Wierdl, Máté},
journal = {Annales de l’institut Fourier},
keywords = {Ergodic theory; recurrence; multiple recurrence; combinatorial additive number theory; ergodic theory; sets of -recurrence; multiple recurrent sets},
language = {eng},
number = {4},
pages = {839-849},
publisher = {Association des Annales de l’institut Fourier},
title = {Sets of $k$-recurrence but not $(k+1)$-recurrence},
url = {http://eudml.org/doc/10173},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Frantzikinakis, Nikos
AU - Lesigne, Emmanuel
AU - Wierdl, Máté
TI - Sets of $k$-recurrence but not $(k+1)$-recurrence
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 4
SP - 839
EP - 849
AB - For every $k\in \mathbb{N}$, we produce a set of integers which is $k$-recurrent but not $(k+1)$-recurrent. This extends a result of Furstenberg who produced a $1$-recurrent set which is not $2$-recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.
LA - eng
KW - Ergodic theory; recurrence; multiple recurrence; combinatorial additive number theory; ergodic theory; sets of -recurrence; multiple recurrent sets
UR - http://eudml.org/doc/10173
ER -

References

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  1. V. Bergelson, Weakly mixing PET, Ergodic Theory Dynamical Systems 7 (1987), 337-349 Zbl0645.28012MR912373
  2. V. Bergelson, Ergodic Ramsey theory-an update, Ergodic Theroy of -actions (1996), 1-61, Cambridge University Press, Cambridge Zbl0846.05095MR1411215
  3. H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 71 (1977), 204-256 Zbl0347.28016MR498471
  4. H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, (1981), Princeton University Press, Princeton, N.J. Zbl0459.28023MR603625
  5. H. Furstenberg, Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1979), 275-291 Zbl0426.28014MR531279
  6. H. Furstenberg, Y. Katznelson, D. Ornstein, The ergodic theoretical proof of Szemerédi’s theorem, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 527-552 Zbl0523.28017MR670131
  7. B. Host, B. Kra, Nonconventional ergodic averages and nilmanifolds, Annals of Math 161 (2005), 397-488 Zbl1077.37002MR2150389
  8. Y. Katznelson, Chromatic numbers of Cayley graphs on ℤ and recurrence, Combinatorica 21 (2001), 211-219 Zbl0981.05038MR1832446
  9. Tamar Ziegler, Universal Characteristic Factors and Furstenberg Averages Zbl1198.37014

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