Let $x\_1,x\_2,\ldots, x\_k \in [0,1]$ be irrational numbers.


I'm interested in what, if anything, can be said about the values $\{nx\_i \bmod 1: n\in \mathbb{N}\}$.
Specifically, I'm interested if there are constraints that we can place on the $x\_i$'s such that for any set of $y\_i$'s and any $\varepsilon \in (0,1/2)$, there exists $n\in \mathbb{N}$ simultaneously satisfying


$$nx\_i- y\_i \bmod 1 \equiv \varepsilon\_i$$
for all $i\in \{1,2,\ldots, k\}$, where all $\varepsilon\_i \in (-\varepsilon, \varepsilon)$.


I know that this is impossible in general. For example, if $x\_1=\sqrt{2},x\_2=2\sqrt{2}$ then we could not have $x\_1 n\bmod 1\approx 1/3$ and $x\_2 n \bmod 1 \approx 1/3$ simultaneously.
However, I maybe if we place some restrictions on $x\_i$, e.g., requiring them all to be pairwise linearly independent over $\mathbb{Q}$, then we could say something of this form?



Continuous variant of the Chinese remainder theorem
Your guess is correct. For $\alpha = (\alpha\_1, \alpha\_2, \ldots, \alpha\_d) \in \mathbb{R}^d$, the values of $m \alpha \bmod 1$ are equidistributed (and in particular dense) in $(\mathbb{R}/\mathbb{Z})^d$ if and only if, for all nonzero $k \in \mathbb{Z}^d$, we have $k \cdot \alpha \not\in \mathbb{Z}$. See, for example, Exercise 5 in [Terry Tao's notes](https://terrytao.wordpress.com/2010/03/28/254b-notes-1-equidistribution-of-polynomial-sequences-in-torii/).


