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 Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 52–70 (Mi cheb453)

Distribution of special algebraic points in domains of small measure

A. G. Gusakova

Institute of Mathematics of the National Academy of Sciences of Belarus

Abstract: Problems related to the distribution of algebraic numbers and points with algebraically conjugate coordinates are a natural generalization of problems connected with estimating of number of integer and rational points in figures and bodies of a Euclidean space.
In this paper we consider a problem related to the distribution of special algebraic points $\boldsymbol{\alpha}=(\alpha_1,\alpha_2)$ with algebraically conjugate coordinates $\alpha_1$ and $\alpha_2$ such that their height and degree are bounded and the absolute values of $P'(\alpha_1)$ and $P'(\alpha_1)$ where $P(t)$ is a minimal polynomial of $\alpha_1$ and $\alpha_2$ are small. The sphere of application of this points is problems related to Mahler's classification of numbers [1] proposed in 1932 and Kosma's classification of numbers [2] proposed some years later. One of this is a question: do Mahler's T-numbers exist? This question has remained unanswered for nearly 40 years and only in 1970 W. Schmidt [3] showed that the class of T-numbers is not empty and proposed the construction of this numbers. Another problem is a question about difference between Mahler's and Koksma's classifications. In 2003 Y. Bugeaud published a paper [4] where he proved that there are exist a numbers with different Mahler's and Koksma's characteristics. Special algebraic points $\boldsymbol{\alpha}=(\alpha_1,\alpha_2)$ considered in this paper are used to prove this results.
We consider special algebraic points $\boldsymbol{\alpha}=(\alpha_1,\alpha_2)$ such that the height of algebraically conjugate numbers $\alpha_1$ and $\alpha_2$ is bounded by $Q$, their degree is bounded by $n$ and $|P'(\alpha_1)|\leq Q^{1-v_1}$, $|P'(\alpha_2)|\leq Q^{1-v_2}$ for $0<v_1,v_2<1$ where $P(t)$ is a minimal polynomial of this numbers. In this paper we obtained the lower and upper bound for the quantity of special algebraic numbers in rectangles with the size of $Q^{-1+v_1+v_2}$.
Bibliography: 22 titles.

Keywords: metric theory of simultaneous Diophantine approximations, Lebesgue measure, conjugate algebraic numbers.

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UDC: 511.42
Accepted:11.03.2016

Citation: A. G. Gusakova, “Distribution of special algebraic points in domains of small measure”, Chebyshevskii Sb., 17:1 (2016), 52–70

Citation in format AMSBIB
\Bibitem{Gus16} \by A.~G.~Gusakova \paper Distribution of special algebraic points in domains of small measure \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 1 \pages 52--70 \mathnet{http://mi.mathnet.ru/cheb453} \elib{https://elibrary.ru/item.asp?id=25795070}