Take as $D$ a complementary $\mathbb{Q}$-vector space of $\mathrm{span}(\{c(n): n\in\mathbb{N}\})$ in $\mathbb{R}$ as $\mathbb{Q}$ vector space. This is not measurable, of course (countably many translate of it covers $\mathbb{R}$ and as you said it can't be of positive measure). It's a version of the Vitali set.