There exist infinitely many positive integers such that...

Question

For any $a_1, a_2, \dots, a_k$, there exist infinitely many positive integers $n$ such that

$$||na_i||$ = $\min{(\{na_i\},1-\{na_i\})}\leq n^{-1/k} \text{ for }1\leq i\leq k$$.

Here, $\{x\}$ denotes the fractional part of $x$.