DITRIBUTION OF SPECIAL ALGEBRAIC POINTS IN DOMAINS OF SMALL MEASURE
https://doi.org/10.22405/2226-8383-2016-17-1-52-70
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 α = (α1, α2) with algebraically conjugate coordinates α1 and α2 such that their height and degree are bounded and the absolute values of P′(α1) and P′(α1) where P(t) is a minimal polynomial of α1 and α2 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 α = (α1, α2) considered in this paper are used to prove this results. We consider special algebraic points α = (α1, α2) such that the height of algebraically conjugate numbers α1 and α2 is bounded by Q, their degree is bounded by n and |P′(α1)| ≤ ≤ Q1−v1 , |P′(α2)| ≤ Q1−v2 for 0 < v1, v2 < 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+v1+v2 .
About the Author
A. G. Gusakova
Институт математики НАН Беларуси
Russian Federation
PhD student, Institute of Mathematics, Belorussian Academy of Sciences
Russian Federation
PhD student, Institute of Mathematics, Belorussian Academy of Sciences
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Review
For citations:
Gusakova A.G. DITRIBUTION OF SPECIAL ALGEBRAIC POINTS IN DOMAINS OF SMALL MEASURE. Chebyshevskii Sbornik. 2016;17(1):52-70. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-1-52-70
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