AI 中文总结
研究整数值非线性多项式有理独立序列，通过研究沿IP的多重遍历多项式平均的特征因子，证明特定集合与有理谱序列生成的IP相交，并推广了Leibman关于沿有理谱IP的多项式平均的逐点收敛定理。
AI 中文摘要
设$p_1,\cdots,p_k$为整数值非线性多项式的有理独立序列。我们证明，对于所有$E\subseteq \mathbb{N}$、每个Folner序列$\Phi$以及每个$\varepsilon>0$，集合$\left\{n\in \mathbb{N}: d_{\Phi}\left(E\cap(E + p_1(n))\cap\cdots\cap (E + p_k(n))\right) > d_{\Phi}(E)^{k + 1}-\varepsilon\right\}$与由有理谱序列生成的每个IP相交。我们的方法涉及研究沿IP的多重遍历多项式平均的特征因子。特别地，我们还证明了沿有理谱IP的多项式平均的逐点收敛定理，推广了Leibman的一个著名结果。
英文摘要
Let $p_1,...,p_k$ be a rationally independent sequence of integer valued nonlinear polynomials. We show that for all $E\subseteq \mathbb{N}$, every Folner sequence $Φ$, and every $\varepsilon>0$, the set $$\left\{n\in \mathbb{N} : d_Φ\left(E\cap(E+p_1(n))\cap\cdots\cap (E+p_k(n))\right) > d_Φ(E)^{k+1}-\varepsilon\right\}$$ intersects every IP generated by a sequence with rational spectrum. Our methods involve the study of the characteristic factors for multiple ergodic polynomial averages along IPs. In particular, we also prove a pointwise convergence theorem for polynomial averages along IPs with rational spectrum, generalizing a well known result of Leibman.
Comments23 pages
