For each $t \in \mathbf{R}$, let $\mathrm{frac}(t)$ be its fractional part.
Question. Fix reals $\alpha_1,\ldots,\alpha_k \in (0,1)$ such that $\sum_{i\le k}\alpha_i<1$. Do there exist infinitely many positive integers $n$ such that $$ \sum_{i\le k}\mathrm{frac}(n\alpha_i)<1\,\,\,\,? $$
The answer is affirmative if $\{\alpha_1,\ldots,\alpha_k\}\setminus \mathbf{Q}$ is a set which is linearly independent over $\mathbf{Q}$ (applying the multidimensional equidistribution theorem). But I don't know in general.