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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 1, Pages 21–42 (Mi izv8215)  

This article is cited in 13 scientific papers (total in 14 papers)

Distribution of real algebraic numbers of arbitrary degree in short intervals

V. I. Bernika, F. Götzeb

a Institute of Mathematics of the National Academy of Sciences of Belarus
b Bielefeld University, Department of Mathematics

Abstract: 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.

DOI: https://doi.org/10.4213/im8215

Full text: PDF file (583 kB)
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English version:
Izvestiya: Mathematics, 2015, 79:1, 18–39

Bibliographic databases:

UDC: 511.42
MSC: 11J83, 11K55
Received: 31.01.2014
Revised: 09.10.2015

Citation: V. I. Bernik, F. Götze, “Distribution of real algebraic numbers of arbitrary degree in short intervals”, Izv. RAN. Ser. Mat., 79:1 (2015), 21–42; Izv. Math., 79:1 (2015), 18–39

Citation in format AMSBIB
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    Remarks

    This publication is cited in the following articles:
    1. N. V. Budarina, F. Götze, “On regular systems of algebraic $p$-adic numbers of arbitrary degree in small cylinders”, Dalnevost. matem. zhurn., 15:2 (2015), 133–155  mathnet  elib
    2. A. S. Kudin, A. V. Lunevich, “Analog teoremy Khinchina v sluchae raskhodimosti v trekhmernom evklidovom prostranstve”, Vestsi Natsyyanalnai akademii navuk Belarusi. Seryya fizika-matematychnykh navuk, 2015, 66–81  elib
    3. A. G. Gusakova, “Raspredelenie spetsialnykh algebraicheskikh tochek v oblastyakh maloi mery”, Chebyshevskii sb., 17:1 (2016), 52–70  mathnet  elib
    4. V. I. Bernik, F. Götze, “Letter to the editors”, Izv. Math., 80:3 (2016), 633–633  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. J. Math. Sci. (N. Y.), 224:2 (2017), 176–198  mathnet  crossref  mathscinet
    6. F. Götze, A. Gusakova, “On algebraic integers in short intervals and near smooth curves”, Acta Arith., 179:3 (2017), 251–265  crossref  mathscinet  zmath  isi  scopus
    7. V. I. Bernik, N. V. Budarina, A. V. Lunevich, Kh. O'Donnel, “Raspredelenie nulei nevyrozhdennykh funktsii na korotkikh otrezkakh”, Chebyshevskii sb., 18:4 (2017), 107–115  mathnet  crossref
    8. V. I. Bernik, A. G. Gusakova, A. S. Kudin, “Otsenki sverkhu i snizu dlya kolichestva algebraicheskikh tochek v korotkikh intervalakh”, Chebyshevskii sb., 18:4 (2017), 116–127  mathnet  crossref
    9. V. Bernik, N. Budarina, H. O'Donnell, “Discriminants of polynomials in the archimedean and non-archimedean metrics”, Acta Math. Hung., 154:2 (2018), 265–278  crossref  mathscinet  zmath  isi  scopus
    10. V. I. Bernik, N. V. Budarina, A. V. Lunevich, Kh. O'Donnell, “Raspredelenie nulei nevyrozhdennykh funktsii na korotkikh otrezkakh II”, Chebyshevskii sb., 19:1 (2018), 5–14  mathnet  crossref  elib
    11. D. V. Vasilev, A. S. Kudin, “Ob otsenkakh sverkhu chisla minimalnykh polinomov s maloi proizvodnoi v korne”, Chebyshevskii sb., 20:2 (2019), 47–54  mathnet  crossref
    12. A. Kudin, D. Vasilyev, “Counting real algebraic numbers with bounded derivative of minimal polynomial”, Int. J. Number Theory, 15:10 (2019), 2223–2239  crossref  mathscinet  zmath  isi
    13. N. I. Kalosha, I. A. Korlyukova, E. V. Guseva, “Intervaly maloi dliny, soderzhaschie algebraicheskoe chislo zadannoi vysoty”, Chebyshevskii sb., 21:1 (2020), 192–199  mathnet  crossref
    14. V N. Budarina , D. Dickinson, V. L. Bernik, “Lower bounds for the number of vectors with algebraic coordinates near smooth surfaces”, Dokl. Nat. Akad. Nauk Belarusi, 64:1 (2020), 7–12  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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