On Diophantine definability and decidability in some infinite totally real extensions of $\mathbb Q$

Let $M$ be a number field, and $W_M$ a set of its non-Archimedean primes. Then let $O_{M,W_M} = \{x \in M\vert \operatorname{ord}_{\mathfrak{t}}x \geq 0, \, \forall \mathfrak{t} \, \not \in W_M\}$. Let $\{p_1,\ldots,p_r\}$ be a finite set of prime numbers. Let $F_{inf}$ be the field generated by all the $p_i^{j}$-th roots of unity as $j \rightarrow \infty$ and $i=1,\ldots,r$. Let $K_{inf}$ be the largest totally real subfield of $F_{inf}$. Then for any $\varepsilon > 0$, there exist a number field $M \subset K_{inf}$, and a set $W_M$ of non-Archimedean primes of $M$ such that $W_M$ has density greater than $1-\varepsilon$, and $\mathbb{Z}$ has a Diophantine definition over the integral closure of $O_{M,W_M}$ in $K_{inf}$.