How to prove that this defined language is regular?

Suppose that $L$ is a regular language and $0<\alpha<1$ and $\alpha \in Q$. Define $L_\alpha$ as

$$L_\alpha = \{\omega \in \Sigma^* \mid \exists \omega_1 \in \Sigma^* .\omega\omega_1 \in L,\frac{|\omega|}{|\omega\omega_1|}=\alpha\}$$

How do I prove that $L_\alpha$ is regular?

Here is my attempt. If we assume that $D$ is the automaton that recognize $L$ with $Q=\{q_0,...,q_n\}$, if $\delta(q_0,\omega\omega_1)\in F$ and $\delta(q_0,\omega)=q_i$ we can define $D_1$ such that $\delta_1(q_0,\omega)=r_{i-1}$ (for $r_i \in F_1)$, but the problem is I can't define $\delta_1(q_0,a)$for $a\in \Sigma$.

Try proving it for $\alpha = 1/2$ first. Construct a non-deterministic automaton that recognizes $L_\alpha$. Before you start reading in the string $w$, guess something about where you will be in the future. What would you like to guess? What would help you?
• Thank you,for other value of $\alpha$(assume that $\omega_1 > \omega$ and their difference is $n$,then we should add n states at the beginning of $D$,am I right? – MAh2014 Sep 21 '17 at 20:44
• @MAh2014, it's your exercise, so I'll let you figure that out. The way to decide if it works is to prove your answer correct. If you're having a hard time with that, try $\alpha = 1/3$ and $\alpha = 2/3$ next. – D.W. Sep 21 '17 at 21:12