Dirichlet's theorem on equidistribtuion tells us that if $\alpha$ is any number, then there are integer multiples of $\alpha$ of arbitrarily small fractional part (well, Dirichlet tells us it for irrationals, and a basic argument works for rationals).
Suppose I have a set $a_1, a_2, ..., a_n$ of real numbers. Given $\epsilon > 0,$ can I find some $N > 0$ so that $Na_1, Na_2, ..., Na_n$ all have fractional part in $[0, \epsilon]$?