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

Let N be a power of $2$ or $3$, and μN the set of N-th roots of unity. We show that the ring of motivic periods of Mixed Tate motives over Z[μN,1/N] 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[μN,1/N] on the motivic fundamental group of Gₘ-μN 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.