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.
@article{IM2_2015_79_1_a1,
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/}
}
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/