Intervals of small measure containing an algebraic number of given height

N. I. Kalosha ; I. A. Korlyukova ; E. V. Guseva

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 213-220

Résumé

Rational numbers are uniformly distributed, even though distances between rational neighbors in a Farey sequence can be quite different. This property doesn't hold for algebraic numbers. In 2013 D. Koleda [6, 7] found the distribution function for real algebraic numbers of an arbitrary degree under their natural ordering. It can be proved that the quantity of real algebraic numbers $ \alpha $ of degree $n$ and height $H( \alpha ) \le Q$ asymptotically equals $c_{1}(n)Q^{n+1}$. Recently it was proved that there exist intervals of length $Q^{- \gamma }, \gamma >1$, free of algebraic numbers $ \alpha , H( \alpha ) \le Q$, however for $0 \le \gamma 1$ there exist at least $c_{2}(n)Q^{n+1- \gamma }$ algebraic numbers in such intervals. In this paper we show that special intervals of length $Q^{- \gamma }$ may contain algebraic numbers even for large values of $ \gamma $, however their quantity doesn't exceed $c_{3}Q^{n+1- \gamma }$. An earlier result by A. Gusakova [16] was proved only for the case $\gamma = \frac{3}{2}$.

Détail

Keywords: algebraic number, Diophantine approximation, uniform distribution, Dirichlet's theorem.