SCHWEITZER 2003 Problem 4

Let $\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty}$ integers such that $a_{n,i}\neq a_{n,j}$ for $1\le i<j\le n , n=2,3,\ldots$ and let $\left\langle y\right\rangle\in [0,1)$ denote the fractional part of the real number $y$ . Show that there exists a real sequence $\{ x_n\}_{n=1}^{\infty}$ such that the numbers $\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle$ are asymptotically uniformly distributed on the interval $[0,1]$ .

(translated by L. Erdős)