TITLE: A property of real numbers concerning integer parts of multiples
QUESTION [11 upvotes]: For a given positive real number $\alpha$, define the set $T_\alpha$ by 
$T_\alpha = \{ [n\alpha] \mid n = 1,2,\dots \}$. What is a necessary and  sufficient condition (in terms of $\alpha$ and $\beta$) to have $T_\alpha \subseteq T_\beta$ ?

REPLY [6 votes]: A partial answer. It is clear that we should consider only $\alpha,\beta>1$. 
Suppose we want to find $n$ such that $n\in T_\beta$, $n\notin T_\alpha$. It is sufficient to find integers $l$ and $k$ such that
$$n\le l\beta<n+1,\qquad k\alpha<n,\quad n+1\le (k+1)\alpha,$$
or equivalently $$\left\{\frac n\beta\right\}\in\{0\}\cup\left(1-\frac 1\beta,1\right),\qquad \left\{\frac n\alpha\right\}\in\left(0,1-\frac 1\alpha\right].$$ If $\alpha$ and $\beta$ are linearly independent with $1$ over $\mathbb{Q}$ then points $\left(\big\{\frac n\beta\big\},\big\{\frac n\alpha\big\}\right)$ are uniformly distributed in $[0,1)^2$ and corresponding $n$ does exist. So the situation $T_\alpha \subseteq T_\beta$ is possible only if $c_1\alpha+c_2\beta+c_3=0$ for some integers $c_1$, $c_2$, $c_3$, $(c_1, c_2, c_3)\ne(0,0,0)$.