We present a new characterization of Muckenhoupt $A_{\infty}$-weights whose logarithm is in $\mathrm{VMO}(\mathbb{R})$ in terms of vanishing Carleson measures on upper-half plane 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.

We propose a representation theorem for $\mathrm{VMO}$ on the unit circle using Carleson's construction and demonstrate a representation theorem for $\mathrm{VMO}$ on $\mathbb{R}^n$ through an iterative process. Additionally, we provide a brand-new characterization of $\mathrm{VMO}$ as an application of our results.