Infinite prime sumsets in structured and $U^k(\mathrm{\Phi })$-uniform sets

By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in $U^k(\mathrm{\Phi })$-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree $k$ of a $U^k(\mathrm{\Phi })$-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 ${\mathrm{\ell }}_1,\cdots ,{\mathrm{\ell }}_k$ 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*}