
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 Tnumbers exist? This question has remained unanswered for nearly 40 years and only in 1970 W. Schmidt [3] showed that the class of Tnumbers 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 $P'(\alpha_1)\leq Q^{1v_1}$, $P'(\alpha_2)\leq Q^{1v_2}$ 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 Received: 20.12.2015 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 5270
\mathnet{http://mi.mathnet.ru/cheb453}
\elib{https://elibrary.ru/item.asp?id=25795070}
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