A motivating study of $W_n$ is given by Grosse-Könne [@Gr]. This overconvergent ring $W_n$ is also the basis (or starting point) of the Monsky-Washnitzer formal cohomology and the rigid cohomology.

More generally, for a growth function $\lambda(x)$, we can also define $$\begin{aligned}
 T_n(\rho, \lambda) &=& \{ \sum_{\mu} a_\mu X^\mu \in k[[X_1, \cdots, X_n]] : |a_\mu|\rho_1^{\lambda^{-1}(\mu_1)}\cdots \rho_n^{\lambda^{-1}(\mu_n)} \stackrel{|\mu| \rightarrow \infty}{\rightarrow} 0 \}.\end{aligned}$$ Similarly, define $$\begin{aligned}
 W_n(\lambda) &=& \bigcup_{\rho\in {\mathbf{R}}^n, \rho_i > 1} T_n(\rho,
 \lambda).\end{aligned}$$ If $\lambda$ is invertible, $W_n(\lambda)$ most closely resembles the ring $k[X; Y, \lambda] = k[X_1, \cdots, X_n; Y, \lambda]$ studied in this paper. If $\lambda(x) = cx$ for some $c > 0$ and all $x$ in ${\mathbf{R}}_{\geq 0}$, then $W_n(\lambda) = W_n$ is the Washnitzer algebra. Similarly, $T_n((1, \cdots, 1), \lambda) = T_n$ for all $\lambda$, and $T_n(\rho, \text{id}) = T_n(\rho)$.

The results of this paper suggest that there may be a $p$-adic cohomology theory for more general growth functions $\lambda(x)$ (other than linear functions), which would help to explain the principal zeroes of Dwork’s unit root zeta function [@Dw], [@Wan2] in the case when $\lambda$ is the exponential function. This is one of the main motivations for the present paper.

[**Acknowledgments**]{}. We would like to thank 