  We present a new characterization of Muckenhoupt $A_{\infty}$-weights whose
logarithm is in $\operatorname{VMO}(\mathbb{R})$ in terms of vanishing Carleson
measures on $\mathbb{R}_+^2$ and vanishing doubling weights on $\mathbb{R}$.
This also gives a novel description of strongly symmetric homeomorphisms on the
real line by using a geometric quantity.
