RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Zap. Nauchn. Sem. POMI: Year: Volume: Issue: Page: Find

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

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).

Full text: PDF file (349 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 511.42
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} 

• http://mi.mathnet.ru/eng/znsl6301
• http://mi.mathnet.ru/eng/znsl/v448/p14

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
•  Number of views: This page: 151 Full text: 37 References: 25