We are dealing with the couples $(\beta,\{b_n\})$, were $\beta$ is a fixed real number and $b_n$ a sequence of real numbers, such that $$ \bigcup_{n=1}^\infty [b_n,\beta b_n)=(0,1) $$ A necessary condition for it is $0<b_n\leq \frac{1}{\beta}<1$ (since $0<b_n<\beta b_n\leq1$). All couples $(\beta,\{\frac{1}{n+\beta-1}\})$, where $\beta>1$ is an integer, have the property.
Now, we are looking for necessary and sufficient conditions for such couples with the property (or their characterizations).
Our first idea is studying the sequences $0<b_n<1$ such that $\limsup\limits_{n\rightarrow \infty} b_n= \frac{1}{\beta}$ and $\liminf\limits_{n\rightarrow \infty} b_n= 0$, or $b_1= \frac{1}{\beta}$ and $\liminf\limits_{n\rightarrow \infty} b_n= 0$, and probably with some other appropriate properties.