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Sets of large values of correlation functions for polynomial cubic configurations

Published online by Cambridge University Press:  19 September 2016

V. BERGELSON  and
A. LEIBMAN
V. BERGELSON
Affiliation:
Department of Mathematics, The Ohio State University, OH 43210, USA email bergelson.1@osu.edu, leibman.1@osu.edu
A. LEIBMAN
Affiliation:
Department of Mathematics, The Ohio State University, OH 43210, USA email bergelson.1@osu.edu, leibman.1@osu.edu
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Abstract

We prove that for any set $E\subseteq \mathbb{Z}$ with upper Banach density $d^{\ast }(E)>0$, the set ‘of cubic configurations’ in $E$ is large in the following sense: for any $k\in \mathbb{N}$ and any $\unicode[STIX]{x1D700}>0$, the set

$$\begin{eqnarray}\displaystyle \biggl\{(n_{1},\ldots ,n_{k})\in \mathbb{Z}^{k}:d^{\ast }\biggl(\mathop{\bigcap }_{e_{1},\ldots ,e_{k}\in \{0,1\}}(E-(e_{1}n_{1}+\cdots +e_{k}n_{k}))\biggr)>d^{\ast }(E)^{2^{k}}-\unicode[STIX]{x1D700}\biggr\} & & \displaystyle \nonumber\end{eqnarray}$$
is an $\text{AVIP}_{0}^{\ast }$-set. We then generalize this result to the case of ‘polynomial cubic configurations’ $e_{1}p_{1}(n)+\cdots +e_{k}p_{k}(n)$, where the polynomials $p_{i}:\mathbb{Z}^{d}\longrightarrow \mathbb{Z}$ are assumed to be sufficiently algebraically independent.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 2 , April 2018 , pp. 499 - 522
Copyright
© Cambridge University Press, 2016 

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