
Intervals of small measure containing an algebraic number of given height
N. I. Kalosha^{a}, I. A. Korlyukova^{b}, E. V. Guseva^{a} ^{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/10.22405/222683832018211213220
Full text:
PDF file (675 kB)
References:
PDF file
HTML file
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 213220
\mathnet{http://mi.mathnet.ru/cheb868}
\crossref{https://doi.org/10.22405/222683832018211213220}
Linking options:
http://mi.mathnet.ru/eng/cheb868 http://mi.mathnet.ru/eng/cheb/v21/i1/p213
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  19  Full text:  6  References:  4 
