I have to prove this fact (found in an article without proof). Let $\alpha \in \mathbb{R}$ be an irrational number. Let $\alpha = [a_0;a_1,a_2,\ldots]$ be the continued fraction expansion. We call $\frac{p_n}{q_n}$ the simplified fraction of the $n$-th convergent, or in other words, $$\frac{p_n}{q_n} = [a_0;a_1,\ldots,a_n].$$ Furthermore we put $\theta_n = q_n\alpha - p_n$. It can be easily proved that $\text{sgn}(\theta_{n}) = (-1)^n$ and that $\left|\theta_n\right| > \left|\theta_{n+1}\right|$.

The theorem that I want to prove states that for every $\beta \in \left]-\alpha,-\alpha +1\right[$ exists and it is unique a sequence of integers $(d_n)_{n\in \mathbb{N}}$ with $$0 \leq d_0 < a_0 \quad \text{and} \quad 0 \leq d_i \leq a_{i+1} \quad \text{ if } i > 0 \quad (1)$$ $$ d_i = a_{i+1} \quad \implies \quad d_{i-1} = 0 \quad \quad \quad \text{if } i > 0 \quad (2)$$ $$ d_{2i} \neq a_{i+1} \quad \text{for infinitely many } i $$ and with the propriety that $$\beta = \sum_{i=0}^{+\infty}\left(d_i\theta_i\right)$$

Note: maybe could be useful to know the proof of Zeckendorf's theorem, which can be found here http://en.wikipedia.org/wiki/Zeckendorf%27s_theorem, since this is a sort of generalization of it.