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A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

Published online by Cambridge University Press:  07 September 2017

JOEL MOREIRA  and
FLORIAN KARL RICHTER
JOEL MOREIRA
Affiliation:
Department of Mathematics, Northwestern University, Evanston, Illinois, USA email joel.moreira@northwestern.edu
FLORIAN KARL RICHTER
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio, USA email richter.109@osu.edu
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Abstract

We investigate how spectral properties of a measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\mathbb{N}\rightarrow \mathbb{N}$, we provide natural conditions on the spectrum $\unicode[STIX]{x1D70E}(T)$ such that, for all $f_{1},\ldots ,f_{k}\in L^{\infty }$,

$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{ja(n)}f_{j}=\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{jn}f_{j}\end{eqnarray}$$
in $L^{2}$-norm. In particular, our results apply to infinite arithmetic progressions,$a(n)=qn+r$, Beatty sequences, $a(n)=\lfloor \unicode[STIX]{x1D703}n+\unicode[STIX]{x1D6FE}\rfloor$, the sequence of squarefree numbers, $a(n)=q_{n}$, and the sequence of prime numbers, $a(n)=p_{n}$. We also obtain a new refinement of Szemerédi’s theorem via Furstenberg’s correspondence principle.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 4 , April 2019 , pp. 1042 - 1070
Copyright
© Cambridge University Press, 2017 

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