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Sets of large values of correlation functions for polynomial cubic configurations

  • V. BERGELSON (a1) and A. LEIBMAN (a1)

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.

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Sets of large values of correlation functions for polynomial cubic configurations

  • V. BERGELSON (a1) and A. LEIBMAN (a1)

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