Distribution of real algebraic numbers of arbitrary degree in short intervals

We consider real algebraic numbers $\alpha$ of degree $\operatorname{deg}\alpha=n$ and height $H=H(\alpha)$. There are intervals $I\subset\mathbb{R}$ of length $|I|$ whose interiors contain no real algebraic numbers $\alpha$ of any degree with $H(\alpha)<\frac12|I|^{-1}$. We prove that one can always find a constant $c_1=c_1(n)$ such that if $Q$ is a positive integer and $Q>c_1|I|^{-1}$, then the interior of $I$ contains at least $c_2(n)Q^{n+1}|I|$ real algebraic numbers $\alpha$ with $\operatorname{deg}\alpha=n$ and $H(\alpha)\le Q$. We use this result to solve a problem of Bugeaud on the regularity of the set of real algebraic numbers in short intervals.