# Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$

By using the LLL algorithm, I tried to find the best simultaneous Diophantine approximation of the three numbers $\sqrt{2}$ and $\sqrt{2 \pm \sqrt{3}}$. I was expecting that to get a precision of $\epsilon$, the common denominator $q$ should be on the order of $\epsilon^{-3}$, because I have three irrational numbers. This is based on a well-known theorem by Dirichlet.

However, it is actually $\epsilon^{-2}$.

The point is that, the extra $\sqrt{2}$ does not make the task harder.

I used the LatticeRudection in Mathematica to find the common q. It turns out that the $q$'s in the triple case coincide with the double case of $\{ \sqrt{2\pm \sqrt{3}} \}$ in many cases!

Can anyone explain this?

If I replace $\sqrt{2}$ by $\sqrt{3}$, it is indeed $\epsilon^{-3}$.

If I do not use $\sqrt{2 \pm \sqrt{3}}$, but the square roots of the prime numbers, indeed the Dirichlet expectation is right.

$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$