Sign intermixing for Riesz bases and frames measured in the Kantorovich-Rubinstein norm

We measure a sign interlacing phenomenon for Bessel sequences $(u_{k})$ in $L^{2}$ spaces in terms of the Kantorovich--Rubinstein mass moving norm $\Vert u_{k}\Vert_{KR}$. Our main observation shows that, quantitatively, the rate of the decreasing $\Vert u_{k}\Vert_{KR}\longrightarrow 0$ havily depends on S. Bernstein $n$-widths of a compact of Lipschitz functions. In particular, it depends on the dimension of the measure space. We have sharp results on the worst and the best rate of convergence of Kantorovich--Rubinstein norms of frames on $d$-dimensional cube. Those rates are sharp.