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In my question here I suggest a possibility for generalization of Furstenberg recurrence theorem needing some hypothesis for that generalization to be hold in the side of convergence of the below average (Multiple recurrence theorem) and existence of positive integer $n$ such that we have positive measure (Furstenberg recurrence theorem)

Furstenberg recurrence theorem: Let $E$ be a subset of a probability space $(X,\mu)$ of positive measure, and let $T: X \to X$ be an invertible measure-preserving shift. Then for any $k \geq 1$ there exists a positive integer $n$ such that $E \cap T^n E \cap T^{2n} E \cap \dots \cap T^{(k-1) n} E$ has positive measure.

Assume we try to generalise this recurrence theorem by the substitution

of linear terms $n,2n,\cdots (k-1)n $ by integer-valued polynomials $p_1(n),p_2(n),\cdots p_i(n),1\leq i \leq n$ of degree ($d \geq 1$), My question here is:

Question: What are the hypothesis we should add to that generalisation to have existence of positive integer $n$ with $E \cap T^{p_1(n)} E \cap T^{p_2(n)} E \cap \dots \cap T^{(k-1) p_i(n)} E$ has a positive measure ?

Probably this question have the same meaning with asking about necessary hypothesis that we should add for the convergence of the lower limit (average) to be hold in $L^2(\mu)$ which is defined by :

$\lim_{N\to \infty} \inf\frac{1}{N} (\sum_{}^{}E \cap T^{p_1(n)} E \cap T^{p_2(n)} E \cap \dots \cap T^{(k-1) p_i(n)} E)>0$ ?

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The extension you want was proved in the 1990s by Bergelson and Leibman. See [1] and also further developments in [2].

[1] Bergelson, Vitaly, and Alexander Leibman. "Polynomial extensions of van der Waerden’s and Szemerédi’s theorems." Journal of the American Mathematical Society 9, no. 3 (1996): 725-753.

[2] Bergelson, Vitaly, Alexander Leibman, and Emmanuel Lesigne. "Intersective polynomials and the polynomial Szemerédi theorem." Advances in Mathematics 219, no. 1 (2008): 369-388.

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