  In this article, we investigate the $\beta$-expansions of real algebraic
numbers. In particular, we give new lower bounds for the number of digit
exchanges in the case where $\beta$ is a Pisot or Salem number. Moreover, we
define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem
numbers, which gives a generalization of Pisot numbers and Salem numbers. Our
method is applicable also to the digit expansions of complex algebraic numbers,
which gives new estimation. In particular, we investigate the digits of
rotational beta expansion by Akiyama and Caalim $[3]$ and zeta-expansion by
Surer $[20]$, where the base is a quasi-Pisot or quasi-Salem number.
