Mixed Tate motives and cyclotomic multiple zeta values of level 2ⁿ or 3ⁿ

Let N be a power of 2 or 3, and ₍ the set of N-th roots of unity. We show that the ring of motivic periods of Mixed Tate motives over Z₍, 1N is spanned by the motivic cyclotomic multiple zeta values of level N. This implies that the action of the motivic Galois group of mixed Tate motives over Z₍, 1N on the motivic fundamental group of G₌-₍ is faithful. This is a generalization of the known results for N\1, 2, 3, 4, 8\ by Deligne and Brown. We also discuss cyclotomic multiple zeta values of weight 2 of other levels.