A sign interlacing phenomenon for Bessel sequences, frames, and Riesz bases
(
u
k
)
(u_{k})
in
L
2
L^2
spaces over the spaces of homogeneous type
Ω
=
(
Ω
,
ρ
,
μ
)
\Omega =(\Omega ,\rho ,\mu )
satisfying the doubling/halving conditions is studied. Under some relations among three basic metric-measure parameters of
Ω
\Omega
, asymptotics is obtained for the mass moving norms
‖
u
k
‖
K
R
\|u_k\|_{KR}
in the sense of Kantorovich–Rubinstein, as well as for the singular numbers of the Lipschitz and Hajlasz–Sobolev embeddings. The main observation shows that, quantitatively, the rate of convergence
‖
u
k
‖
K
R
→
0
\|u_k\|_{KR}\to 0
mostly depends on the Bernstein–Kolmogorov
n
n
-widths of a certain compact set of Lipschitz functions, and the widths themselves mostly depend on the interplay between geometric doubling and measure doubling/halving numerical parameters. The “more homogeneous” is the space, the sharper are the results.