Lower bounds for the number of vectors with algebraic coordinates near smooth surfaces
https://doi.org/10.29235/1561-8323-2020-64-1-7-12
Abstract
Let z = f(x, y) be a surface in three-dimensional Euclidean space. Consider a neighborhood V of this surface, whose points satisfy the inequality | f(x, y) - z| < Q -Y, where 0 < у < 1 and Q is a sufficiently large positive integer. In the works of Huxley, Beresnevich, Velani, the distribution of rational points in V has been started. In this article, we study the distribution of points with real conjugate algebraic coordinates ᾱ = α1α2α3 in V. For some c1 = c1(n), a lower bound is obtained in the form of c2 Q n+1-Y for the number of algebraic numbers of degree n ≥ 3 and of height at most c3 Q.
About the Authors
N. V. Budarina
Hundalk Institute of Technology
Ireland
Ireland
Budarina Nataliya V. - D. Sc. (Physics and Mathematics).
A91 К584, Dublin Road, Dundalk
D. Dickinson
National University of Ireland
Ireland
Ireland
Dickinson Detta - Ph. D.
Maynooth
V. Bernik
Institute of Mathematics, National Academy of Sciences of Belarus
Belarus
Belarus
Bernik Vasiliy I. - D. Sc. (Physics and Mathematics), Professor, Chief researcher.
11, Surganov Str., 220072, Minsk
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