What subsets of rationals have been defined where each element equals the sum of a sequence of numbers?

Question

In particular, I'm interested in the rationals that result from adding a finite sequence of consecutive integer powers of two. Has it been studied somewhere?

Update
This formalized example might clarify the point:
$$\left\{\frac{a}{b}\in\mathbb{Q}\mathrel{}\middle|\mathrel{} \frac{a}{b}=\sum_{x\in X} x,X \subset \mathscr P(\mathbb{R})\right\}$$ Can't provide concrete numeric examples so far.