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. 



Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$
$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$


