Counting algebraic numbers in short intervals with rational points

  • Vasily I. Bernik Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus
  • Friedrich Götze Bielefeld University, 25 Universitätsstraße, Bielefeld D­33615, Germany
  • Nikolai I. Kalosha Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus
DOI: https://doi.org/10.33581/2520-6508-2019-1-4-11

Abstract

In 2012 it was proved that real algebraic numbers follow a non­uniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small denominators. In this article the authors are first to specify a gene ral condition that guarantees the presence of a large quantity of real algebraic numbers in a small interval. Under this condition, the distribution of real algebraic numbers attains even stronger regularity properties, indicating that there is a chance of proving Wirsing’s conjecture on approximation of real numbers by algebraic numbers and algebraic integers.

Author Biographies

Vasily I. Bernik, Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus

doctor of science (physics and mathematics), full professor; chief researcher at the department of number theory

Friedrich Götze, Bielefeld University, 25 Universitätsstraße, Bielefeld D­33615, Germany

doctor of science (mathematics), full professor

Nikolai I. Kalosha, Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus

PhD (physics and mathematics); researcher at the department of number theory

References

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Published
2019-04-08
Keywords: algebraic number, Diophantine approximation, uniform distribution, Dirichlet’s theorem, Khinchine’s theorem
How to Cite
Bernik, V. I., Götze, F., & Kalosha, N. I. (2019). Counting algebraic numbers in short intervals with rational points. Journal of the Belarusian State University. Mathematics and Informatics, 1, 4-11. https://doi.org/10.33581/2520-6508-2019-1-4-11
Issue
No 1 (2019): Journal of the Belarusian State University. Mathematics and Informatics
Section
Mathematical Logic, Algebra and Number Theory

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