such that $$\label{eq:estr}
  m(\omega_d(G))\leq r_0(G)\leq |G_0|\leq 2^{\omega_d(G)}.$$

Since every pseudocompact torsion group must be of bounded order, the theorem is trivial (and vacuous) for such groups, we may assume that $G$ is nontorsion.

Decompose $G$ as in Lemma \[descomp\]: $$\left( \bigoplus_{p^k \in {{\mathbb P}}_0^\uparrow} \bigoplus_{\gamma(p^k)}{{\mathbb Z}}(p^k) \right)\bigoplus H$$ with ${{\mathbb P}}_0^{\uparrow}$ a finite subset of ${{\mathbb P}}^\uparrow$ and $$\left| n H\right|=|H| \mbox{ for all }n\in {{\mathbb N}}.$$

Split ${{\mathbb P}}_0^\uparrow={{\mathbb P}}_1^\uparrow\cup {{\mathbb P}}_2^\uparrow$ as in Definition \[def:split\] and define $$G_0=\bigoplus_{p_i^{k_i} \in {{\mathbb P}}_2^\uparrow}
\bigoplus_{\gamma(p_i^{k_i})}{{\mathbb Z}}(p_i^{k_i})\bigoplus H.$$ We will prove that the inequalities \[eq:estr\] hold for $w_d(G)=w(nG_0)$.

Lemma \[estrdikgio\] proves that there is some $n\in {{\mathbb N}}$ with $$\label{inwd} m(w(nG_0))\leq r_0(G_0)\leq
 2^{w(nG_0)}.$$ If $|G_0|=\gamma(p_i^{k_i})$ for some $p_i^{k_i} \in {{\mathbb P}}_2^\uparrow$, it follows from the definition of $P_2^\uparrow$ that $|G_0|=
r_0(G)$ and is deduced from . If, otherwise, $|G_0|=|H|$, then $|nG_0|\geq |nH|=|H|= |G_0|$ and we deduce that $|G_0|=|nG_0|$ and thus that $|G_0|\leq 2^{w(nG_0)}$. This together with gives again with $w_d(G)=w(nG_0)$.

The cardinal $w_d(G)$ used in Theorem \[estrpseu\] is precisely the *divisible weight* of $G$ that was introduced and studied by Dikranjan and Giordano-Bruno [@dikrgior08]. We refer the reader to that paper to get an idea of the important role played by the 