On $q$-ary periodic sequences

We consider the problem of estimating the possible number of periods and the length of the periodic part of an irrational number depending on its measure of irrationality $\beta$. We state that the expansion of the fractional part of an irrational number $\alpha$ cannot start from the nonperiodic part of length $(1-\delta)N$ and end with the periodic part of the length $\delta N$, regardless of the numeral system.