Suppose we have $n$ irrational numbers $\{ x_1, x_2, \ldots, x_n \}$. For a generic set of such numbers, we have the theorem that there exist infinitely many integers $q$ such that

$$ \max\{ \parallel q x_1 \parallel, \parallel q x_2 \parallel, \ldots, \parallel q x_n \parallel\} < q^{-1/n}.$$

Here $\parallel \cdot \parallel $ means the distance of the number to the closest integer.

The point is, could this theorem be strengthened if the $n$ numbers are somehow related? As an example, what if

$$ \{x_1, x_2, x_3, x_4 \} = \{ \sqrt{2}, \sqrt{3}, \sqrt{2+\sqrt{3}}, \sqrt{2-\sqrt{3}}\}?$$

Suppose we have $n$ irrational numbers $\{ x_1, x_2, \ldots, x_n \}$. For a generic set of such numbers, we have the well-known theorem that there exist infinitely many integers $q$ such that

$$ \max\{ \parallel q x_1 \parallel, \parallel q x_2 \parallel, \ldots, \parallel q x_n \parallel\} < q^{-1/n}.$$

Here $\parallel \cdot \parallel $ means the distance of the number to the closest integer.

The point is, could this theorem be strengthened if the $n$ numbers are somehow related? As an example, what if

$$ \{x_1, x_2, x_3, x_4 \} = \{ \sqrt{2}, \sqrt{3}, \sqrt{2+\sqrt{3}}, \sqrt{2-\sqrt{3}}\}?$$