Let $H$ be a $\mathbb{Q}$-basis of $\mathbb{R}$ (Hamel basis).
It is known, that $H$ is dense in some interval, also is measurable with measure $0$ or nonmeasurable with infinite outer measure.
Let $H_1,H_2$ be two Hamel bases and $D=H_1-H_2=\{h_1-h_2\colon h_1\in H_1, h_2\in H_2\}$.
Are there are any known facts about the measure of $D$ - e.g. $D$ is nonmeasurable or contains an interval or maybe has infinite outer measure?