Geodesic
Distribution of real algebraic numbers of arbitrary degree in short intervals
Izvestiya. Mathematics , Tome 79 (2015) no. 1, pp. 18-39

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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.
Keywords: algebraic numbers, regular systems. 
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     author = {V. I. Bernik and F. G\"otze},
     title = {Distribution of real algebraic numbers of arbitrary degree in short intervals},
     journal = {Izvestiya. Mathematics },
     pages = {18--39},
     publisher = {mathdoc},
     volume = {79},
     number = {1},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2015_79_1_a1/}
}
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V. I. Bernik; F. Götze. Distribution of real algebraic numbers of arbitrary degree in short intervals. Izvestiya. Mathematics , Tome 79 (2015) no. 1, pp. 18-39. http://geodesic.mathdoc.fr/item/IM2_2015_79_1_a1/