Infinite prime sumsets in structured and $U^k(Φ)$-uniform sets

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 co