Distribution of special algebraic points in domains of small measure

A. G. Gusakova

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 52-70

Résumé

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 $0$ 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.

Détail

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