In continuation of Matsumoto's paper (Nonlinearity 25:1495-1511, 2012) we show that various subspaces are -dense in the space of orientation-preserving -diffeomorphisms of the circle with rotation number , where is any prescribed Liouville number. In particular, for every odometer of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to O.