Simultaneous Diophantine approximation in the non-generic case

Question

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

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

Answer 1

Not sure exactly what you have in mind for a strengthening, but for the example you give I believe you can find infinitely many integers $q$ with $$ \max\{ \parallel q x_1 \parallel, \parallel q x_2 \parallel, \ldots, \parallel q x_4 \parallel\} < q^{-1/3}.$$

Here is a plot of $q^{1/3}\parallel q \sqrt 2 \parallel$ vs. $q^{1/3}\parallel q \sqrt 3 \parallel$ for 20,000 or so good $q$'s that satisfy the original inequality:

Zooming in a bit we get a bit more interesting picture:

So I would think it is quite plausible that the theorem can sometimes be strengthened when the $n$ numbers are related.