Geodesic
Dynamical systems with low recurrence rate
Sbornik. Mathematics, Tome 197 (2006) no. 11, pp. 1697-1712

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The question on the recurrence rate of a dynamical system in a metric space of finite Hausdorff measure is considered. For such systems upper bounds for the rate of simple recurrence are due to Boshernitzan and ones for the rate of multiple recurrence to the present author. The subject of the paper are lower bounds for the rate of multiple recurrence. More precisely, an example of a dynamical system (an odometer or a von Neumann transformation) with a low rate of multiple recurrence is constructed. Behrend's theorem on sets containing no arithmetic progressions is used in the proof. Bibliography: 22 titles. 
@article{SM_2006_197_11_a6,
     author = {I. D. Shkredov},
     title = {Dynamical systems with low recurrence rate},
     journal = {Sbornik. Mathematics},
     pages = {1697--1712},
     publisher = {mathdoc},
     volume = {197},
     number = {11},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2006_197_11_a6/}
}
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I. D. Shkredov. Dynamical systems with low recurrence rate. Sbornik. Mathematics, Tome 197 (2006) no. 11, pp. 1697-1712. http://geodesic.mathdoc.fr/item/SM_2006_197_11_a6/