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

The regionally proximal relation of order $d$ along arithmetic progressions, namely ${\mathbf{A}\mathbf{P}}^{[d]}$ for $d\in \mathbb{N}$, is introduced and investigated. It turns out that if $(X,T)$ is a topological dynamical system with ${\bf AP}^{[d]}=\Delta$,theneachergodicmeasureof$(X,T)$isisomorphictoa$d$-steppro-nilsystem,andthus$(X,T)$ 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 ${\mathbf{A}\mathbf{P}}^{[d]}={\mathbf{R}\mathbf{P}}^{[d]}$ for each $d\in \mathbb{N}$. It follows that for a minimal $\mathrm{\infty }$-pro-nilsystem, ${\bf AP}^{[d]}={\bf RP}^{[d]}$foreach$d\in {\mathbb N}$. 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.