Let $\alpha$ be an irrational real number. I wonder whether $\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1] in which $\{x\}$ means the fractional part of x.
This is equivalent to the following: $$\forall \alpha\in(0,1)\setminus\mathbb{Q},E_j=\mathrm{closure~of ~}\left\{\{j^n\alpha\}:n\in\mathbb{N}\right\},j=2,3,\mathrm{~then~}E_2+E_3=[0,1].$$
In the proof of the case $2^n3^m\alpha$, the semi-group property of $\{2^n3^m:n,m\in\mathbb{N}\}$ is used. But this property is missing in the above question.
Any hint would be appreciated!