Imposing psendocompact group topologies on Abeliau groups

The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, ${\displaystyle m(\alpha )\le 2^{\alpha }}$. We show:    Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m${\displaystyle (\alpha )\le r_0\left(G\right)\le \gamma \le 2^{\alpha }}$, or α > ω and ${\displaystyle {\alpha }^{\omega }\le r_0\left(G\right)\le 2^{\alpha }}$, then G admits a pseudocompact group topology of weight α.  Theorem 4.15. Every connected, pseudocompact Abelian group G with wG = α ≥ ω satisfies ${\displaystyle r_0\left(G\right)\ge m(\alpha )}$.  Theorem 5.2(b). If G is divisible Abelian with ${\displaystyle 2^{r_0\left(G\right)}\le \gamma }$, then G admits at most ${\displaystyle 2^{\gamma }}$-many pseudocompact group topologies.  Theorem 6.2. Let ${\displaystyle \beta ={\alpha }^{\omega }}$ or ${\displaystyle \beta =2^{\alpha }}$ with β ≥ α, and let ${\displaystyle \beta \le \gamma <\kappa \le 2^{\beta }}$. Then both ${\displaystyle {\oplus }_{\gamma }\mathbb{Q}}$ and the free Abelian group on γ-many generators admit exactly ${\displaystyle 2^{\kappa }}$-many pseudocompact group topologies of weight κ. Of these, some ${\displaystyle {\kappa }^{+}}$-many form a chain and some ${\displaystyle 2^{\kappa }}$-many form an anti-chain.