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

中文翻译：

---

β 扩展的数字频率

令 $$\beta >1$$ β > 1 为非整数。首先，我们证明 Lebesgue 几乎每个数字都有给定频率的 $$\beta $$ β -展开当且仅当 Lebesgue 几乎每个数字都有无限多个相同给定频率的 $$\beta $$ β -展开。然后我们推导出 Lebesgue 几乎每个数都有无穷多个平衡的 $$\beta $$ β -展开，其中有限字母表上的无限序列 $$\{0,1 , \ldots ,m\}$$ { 0 , 1 , ... , m } 如果数字 $$k$$ k 的频率等于数字 $$mk$$ m - k 对于所有 $$k\in \{0,1 , \ ldots ,m\}$$ k ∈ { 0 , 1 , … , m } 。最后我们考虑变频并证明对于每个伪黄金比率 $$\beta \in (1,2)$$ β ∈ ( 1 , 2 ) ，存在一个常数 $$c=c(\beta )>0 $$ c = c ( β ) > 0 使得对于任何 $$p\in [\frac{1}{2}-c,