Is the following possible ? $$ Th( \mathbb{Q}, +, \leq ) \leq^{\log}_m Th( \mathbb{N}, +, \leq )$$
I believe it is not possible since Natural Numbers are not dense. It is also not possible $$ Th( \mathbb{N}, +, \leq ) \leq^{\log}_m Th( \mathbb{Q}, +, \leq )$$ since complexity of $Th( \mathbb{N}, +, \leq )$ is much higher than of $Th( \mathbb{R}, +, \leq )$
But I do not feel extreme confidence in my statement and "Googling" does not help to come up with anything more comprehensive than that.