Let $A$ be an unbounded set in $\mathbb{R}$. Then consider the set $M(t)=\{a\mod t|a \in A \}$ in an interval $[0,t]$,for given $t>0$. Then some of $t$ will make $M(t)$dense in $[0,t]$. (This is my conjecture, so not maybe...) But my question is : can we always find $t$ that $M(t)$ dense, that is arbitrary large? or, is the set $\{t>0|M(t)$ is dense $\}$ is unbounded or not? I think this may be true..
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