Hamel Spaces and Distal Expansions

In this note, we construct a distal expansion for the structure $($\mathbb{R}$; +,<,H)$,where$H\subseteq $\mathbb{R}$$isadense$$\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.