Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems

Abstract: The regionally proximal relation of order $d$ along arithmetic progressions, namely ${\bf AP}^{[d]}$ for $d\in \N$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${\bf AP}^{[d]}=\Delta$, then each ergodic measure of $(X,T)$ is isomorphic to a $d$-step pro-nilsystem, and thus $(X,T)$ has zero entropy. Moreover, it is shown that if $(X,T)$ is a strictly ergodic distal system with the property that the maximal topological and measurable $d$-step pro-nilsystems are isomorphic, then ${\bf AP}^{[d]}={\bf RP}^{[d]}$ for each $d\in {\mathbb N}$. It follows that for a minimal $\infty$-pro-nilsystem, ${\bf AP}^{[d]}={\bf RP}^{[d]}$ for each $d\in {\mathbb N}$. An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.