I am interested in a statement of the following form.
Let $K$ be a real number field and consider real numbers $(\xi^-, \xi^+)$.
Assume that there are infinitely many pairs of real algebraic numbers $(\alpha^-, \alpha^+)$
which are quadratic conjugates over $K$ (but maybe all lying in different quadratic extensions),
and positive real numbers $\epsilon, c$ such that
$$|(\xi^+ - \alpha^+ )(\xi^- - \alpha^-)| \le c/disc(\alpha)^{2+\epsilon}$$
where $disc(\alpha)$ is the discriminant of a primitive quadratic form
with coefficients in $O_K$ vanishing at $(\alpha^-, \alpha^+)$.
The conclusion is that:
- either both $\xi^\pm$ lie in an at most quadratic extension of $K$,
- or else at least one of $\xi^\pm$ is transcendental.
I am willing to assume (and i think it is needed) that among all conjugates of $\alpha^\pm$ over $Q$,
the maximal modulus is achieved by $max(|\alpha^- |, |\alpha^+|)$.
This hypothesis is not necessary when $K=Q$ which is already of interest.
I am aware of a couple of results by K. Mahler and W. Schmidt, such as those cited or proven in
Simultaneous approximation to algebraic numbers by elements of a number field, by W. Schmidt
https://link.springer.com/article/10.1007/BF01533775
but the closest i found always assumed that the $\alpha$ all lied in a given number field.
All i assume is that they are quadratic over $K$, in particular their degree is bounded.
One can formulate this in terms of the approximation of real quadratic forms
by quadratic forms whose coefficients lie in the fixed number field $K$,
or equivalently of elements in $Proj(sl(2,R))$ by elements in $Proj(sl(2,K))$,
which may fit into the framework of "Diophantine approximation in algebraic groups".
