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Zap. Nauchn. Sem. POMI, 2016, Volume 448, Pages 14–47 (Mi znsl6301)  

This article is cited in 2 scientific papers (total in 2 papers)

On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves

V. Bernika, F. Götzeb, A. Gusakovaa

a Institute of Mathematics of the National Academy of Sciences of Belarus, Surganov str. 11, Minsk 220072, Belarus
b Department of Mathematics, University of Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

Abstract: Let $\varphi\colon\mathbb R\to\mathbb R$ be a continuously differentiable function on a finite interval $J\subset\mathbb R$, and let $\boldsymbol\alpha=(\alpha_1,\alpha_2)$ be a point with algebraically conjugate coordinates such that the minimal polynomial $P$ of $\alpha_1,\alpha_2$ is of degree $\leq n$ and height $\leq Q$. Denote by $M^n_\varphi(Q,\gamma,J)$ the set of points $\boldsymbol\alpha$ such that $|\varphi(\alpha_1)-\alpha_2|\leq c_1Q^{-\gamma}$. We show that for $0<\gamma<1$ and any sufficiently large $Q$ there exist positive values $c_2<c_3$, where $c_i=c_i(n)$, $i=1,2$, that are independent of $Q$ and such that $c_2\cdot Q^{n+1-\gamma}<# M^n_\varphi(Q,\gamma,J)<c_3\cdot Q^{n+1-\gamma}$.

Key words and phrases: algebraic numbers, metric theory of Diophantine approximation, Lebesgue measure.

Funding Agency Grant Number
Universität Bielefeld SFB-701
Supported by SFB-701, Bielefeld University (Germany).


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English version:
Journal of Mathematical Sciences (New York), 2017, 224:2, 176–198

Bibliographic databases:

UDC: 511.42
Received: 25.10.2016
Language:

Citation: V. Bernik, F. Götze, A. Gusakova, “On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves”, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Zap. Nauchn. Sem. POMI, 448, POMI, St. Petersburg, 2016, 14–47; J. Math. Sci. (N. Y.), 224:2 (2017), 176–198

Citation in format AMSBIB
\Bibitem{BerGotGus16}
\by V.~Bernik, F.~G\"otze, A.~Gusakova
\paper On the distribution of points with algebraically conjugate coordinates in a~neighborhood of smooth curves
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXVII
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 448
\pages 14--47
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6301}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3576247}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2017
\vol 224
\issue 2
\pages 176--198
\crossref{https://doi.org/10.1007/s10958-017-3404-6}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85019764176}


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    This publication is cited in the following articles:
    1. 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
    2. V. I. Bernik, N. V. Budarina, H. O'Donnell, A. V. Lunevich, “Raspredelenie nulei nevyrozhdennykh funktsii na korotkikh otrezkakh II”, Chebyshevskii sb., 19:1 (2018), 5–14  mathnet  crossref  elib
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