As the title describes, I will post here my question clearer:
Let $z=\frac{m}{2^k}$ be a dyadic rational number in $(0,1)$ where $m$ is odd and $k >0$, and also $n$ is a fixed positive integer. Let's denote $N(z,n)$ to be the number of $n$ tuples $(i_1,...,i_n)$ of positive integers such that $z=\sum_{j=1}^{n} \frac{1}{2^{i_j}}$ (I assume $i_1 \leq i_2 ... \leq i_n$).
My question is : Are these numbers $N(z,n)$ always finite? If yes, I want to have an upper bound for this $N(z,n)$ ... Thanks !