  Let $\beta>1$ be a non-integer. First we show that Lebesgue almost every
number has a $\beta$-expansion of a given frequency if and only if Lebesgue
almost every number has infinitely many $\beta$-expansions of the same given
frequency. Then we deduce that Lebesgue almost every number has infinitely many
balanced $\beta$-expansions, where an infinite sequence on the finite alphabet
$\{0,1,\cdots,m\}$ is called balanced if the frequency of the digit $k$ is
equal to the frequency of the digit $m-k$ for all $k\in\{0,1,\cdots,m\}$.
Finally we consider variable frequency and prove that for every pseudo-golden
ratio $\beta\in(1,2)$, there exists a constant $c=c(\beta)>0$ such that for any
$p\in[\frac{1}{2}-c,\frac{1}{2}+c]$, Lebesgue almost every $x$ has infinitely
many $\beta$-expansions with frequency of zeros equal to $p$.
