# Multiple ergodic theorems for arithmetic sets

Abstract : We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemer\'edi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter $n$ to those integers that have an even number of distinct prime factors, or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analysed using a recent structural result for bounded multiplicative functions proved by the authors.
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Journal articles

https://hal-upec-upem.archives-ouvertes.fr/hal-01267077
Contributor : Bernard Host <>
Submitted on : Wednesday, February 3, 2016 - 7:56:02 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

### Citation

Nikos Frantzikinakis, Bernard Host. Multiple ergodic theorems for arithmetic sets. Transactions American Mathematical Society, American Mathematical Society, 2017, 369 (10), pp.7085-7105. ⟨10.1090/tran/6870⟩. ⟨hal-01267077⟩

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