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 Chebyshevskii Sb., 2020, Volume 21, Issue 1, Pages 213–220 (Mi cheb868)

Intervals of small measure containing an algebraic number of given height

N. I. Kaloshaa, I. A. Korlyukovab, E. V. Gusevaa

a Institute of mathematics of the National Academy of Sciences of Belarus (Minsk)
b Yanka Kupala State University of Grodno

Abstract: 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 $\gamma = \frac{3}{2}$.

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

DOI: https://doi.org/10.22405/2226-8383-2018-21-1-213-220

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UDC: 511.42

Citation: N. I. Kalosha, I. A. Korlyukova, E. V. Guseva, “Intervals of small measure containing an algebraic number of given height”, Chebyshevskii Sb., 21:1 (2020), 213–220

Citation in format AMSBIB
\Bibitem{KalKorGus20} \by N.~I.~Kalosha, I.~A.~Korlyukova, E.~V.~Guseva \paper Intervals of small measure containing an algebraic number of given height \jour Chebyshevskii Sb. \yr 2020 \vol 21 \issue 1 \pages 213--220 \mathnet{http://mi.mathnet.ru/cheb868} \crossref{https://doi.org/10.22405/2226-8383-2018-21-1-213-220}