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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

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

This article is cited in 5 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.

UDC: 511.42

Received: 25.10.2016

Language: English


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

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