The regionally proximal relation of order d along arithmetic progressions,namely AP[d]for d ∈ N,is introduced and investigated.It turns out that if (X,T) is a topological dynamical system with AP[d]=A,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 AP[d]=RP[d]for each d ∈ N.It follows that for a minimal ∞-pro-nilsystem,AP[d]=RP[d]for each d ∈ 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.