  This paper provides a necessary and sufficient condition on Tauberian
constants associated to a centered translation invariant differentiation basis
so that the basis is a density basis. More precisely, given $x \in
\mathbb{R}^n$, let $\mathcal{B} = \cup_{x \in \mathbb{R}^n} \mathcal{B}(x)$ be
a collection of bounded open sets in $\mathbb{R}^n$ containing $x$. Suppose
moreover that these collections are translation invariant in the sense that,
for any two points $x$ and $y$ in $\mathbb{R}^n$ we have that $\mathcal{B}(x +
y) = \{R + y : R \in \mathcal{B}(x)\}.$ Associated to these collections is a
maximal operator $M_{\mathcal{B}}$ given by $$M_{\mathcal{B}}f(x) :=\sup_{R \in
\mathcal{B}(x)} \frac{1}{|R|} \int_R |f|.$$ The Tauberian constants
$C_{\mathcal{B}}(\alpha)$ associated to $M_{\mathcal{B}}$ are given by
$$C_{\mathcal{B}}(\alpha) :=\sup_{E \subset \mathbb{R}^n \atop 0 < |E| <
\infty} \frac{1}{|E|}|\{x \in \mathbb{R}^n :\, M_{\mathcal{B}}\chi_E(x) >
\alpha\}|.$$ Given $0 < r < \infty$, we set $\mathcal{B}_r(x) :=\{R \in
\mathcal{B}(x) : \mathrm{diam } R < r\}$, and let $\mathcal{B}_r :=\cup_{x \in
\mathbb{R}^n} \mathcal{B}_r (x).$ We prove that $\mathcal{B}$ is a density
basis if and only if, given $0 < \alpha < \infty$, there exists $ r = r(\alpha)
>0$ such that $C_{\mathcal{B}_r}(\alpha) < \infty$. Subsequently, we construct
a centered translation invariant density basis $\mathcal{B} = \cup_{x \in
\mathbb{R}^n} \mathcal{B}(x)$ such that there does not exist any $0 < r$
satisfying $C_{\mathcal{B}_{r}}(\alpha) < \infty$ for all $0 < \alpha < 1$.
