3 problems

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a strictly increasing sequence. For a system $(X,{\mathcal X},\mu,T)$, say that $a$ is good for seminorm control along $\ell$-term arith…

Let $(G/\Gamma,\mu_{G/\Gamma},R_\alpha)$ be an ergodic nilsystem, and let $k_1,\ldots,k_\ell\in\mathbb Z$ be nonzero and distinct. For functions…

Let $a\colon\mathbb N\to\mathbb Z$ be strictly increasing. The notions of being good for seminorm control and for degree-$(\ell+1)$ seminorm control along $\ell$-term arithmetic pr…