Regionally proximal relation of order d along arithmetic progressions and nilsystems

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] = Δ, 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.