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

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