Abstract In [2], Huayi Chen introduced approximable graded algebras, which he uses to prove a Fujita-type theorem in the arithmetic setting, and asked if any such algebra is the graded ring of a big line bundle on a projective variety. This was proved to be false in [ 8]. Continuing the analysis started in [8], we show that while not every approximable graded algebra is a sub algebra of the section ring of a big line bundle, we can associate to any approximable graded algebra $\textbf{B}$ a projective variety $X(\textbf{B})$ and an infinite divisor $D(\textbf{B}) =\sum _{i=1}^\infty a_i D_i$ with $a_i\rightarrow 0$ such that $\textbf{B}$ is a subalgebra of $$\begin{equation*} R( D(\textbf{B}))=\oplus_n H^0(X(\textbf{B}), n D(\textbf{B})).\end{equation*}$$We also establish a partial converse to these results by showing that if an infinite divisor $D=\sum _i a_iD_i$ converges in the space of numerical classes, then any full-dimensional sub-graded algebra of $\oplus _mH^0(X, \lfloor mD \rfloor ))$ is approximable.