Hamel Spaces and Distal Expansions

In this note, we construct a distal expansion for the structure $(\mathbb{R}; +,<,H)$, where $H\subseteq \mathbb{R}$ is a dense $\mathbb{Q}$-vector space basis of $\mathbb{R}$ (a so-called Hamel basis). Our construction is also an expansion of the dense pair $(\mathbb{R}; +,<,\mathbb{Q})$ and has full quantifier elimination in a natural language.