Let $x_1,x_2,\ldots,x_n > 0$ be such that $x_1+x_2+\cdots+x_n < 1$. Is it true that there must exist a positive integer $k$ such that $$\{x_1k\}+\{x_2k\}+\cdots+\{x_nk\} > n-1?$$


This looks closely related to the [density of the fractional part](https://math.stackexchange.com/questions/903142/for-an-irrational-number-a-the-fractional-part-of-na-for-n-in-mathbb-n-is). The case $n=1$ is obvious, while $n=2$ appeared in this [question](https://math.stackexchange.com/questions/3664348/pouring-water-from-bottles).

Note that if we allow $x_1+x_2+\cdots+x_n=1$, the statement no longer holds, e.g. by taking $x_1=x_2=\cdots=1/n$.