Consider a basis $B$ of the vector space $\Bbb{R}$ over the field $\Bbb{Q}$, known as a Hamel basis.
Is $B$ necessarily dense in $\Bbb{R}$?
It seems hard to answer this question since we can't actually construct a Hamel basis and I don't know any of its properties other than that it exists!
To some extent, the opposite is true. Given any positive real number $M$, there exists a $\mathbb Q$-basis of $\mathbb R$, which avoids the interval $(-M,M)$.
Because if there is a basis element in this interval, you can just replace it by a sufficiently large rational multiple to get out of this interval.
With basically the same argument, you get that for any two real numbers $a<b$, there exists a $\mathbb Q$-basis of $\mathbb R$, which is contained in $(a,b)$. So the elements of a basis can be concentrated in an arbitrary small interval.