AI 中文总结
该论文通过新的遍历理论技术，确定了$U^k(\Phi)$-一致集合和Nil-Bohr集合中的无限集和配置，解决了相关猜想，并证明了平移素数构成的无限集和及其在多项式下的模1分布性质。
AI 中文摘要
通过引入nilsystems中新的遍历理论技术，我们确定了哪些无限集和配置出现在$U^k(\Phi)$-一致集合和Nil-Bohr集合中。更确切地说，我们的第一个结果将$U^k(\Phi)$-一致集合的度数$k$与其包含的集和的多样性联系起来，解决了Kra、Moreira、Richter和Robertson的一个猜想。限制在Nil-Bohr集合上，我们证明了存在以平移素数$\mathbb{P}-1$为加数的无限集和。作为推论，我们证明了对任何具有度数$k$的首项无理系数的实多项式$Q(n)$，以及任何自然数$\ell_1, \cdots, \ell_k$，存在一个无限集合$P\subset \mathbb{P}$，使得对于所有$I \subset P, |I| = \ell_1, \ldots, \ell_k$，有$Q\Big(\sum_{p \in I} p\Big) \in U \pmod 1$。
英文摘要
By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in $U^k(Φ)$-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree $k$ of a $U^k(Φ)$-uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes $\mathbb{P}-1$. As a consequence, we show that for any real polynomial $Q(n)$ with leading irrational coefficient of degree $k$, and any natural numbers $\ell_1, \cdots, \ell_k$ there is an infinite set $P\subset \mathbb{P}$ such that \begin{equation*} Q\Big(\sum_{p \in I} p\Big) \in U \pmod 1 \quad \text{ for all } I \subset P , |I| = \ell_1, \ldots, \ell_k. \end{equation*}
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