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 = α 1 α 2 α 3 in V. For some c 1 = c 1 ( n ), a lower bound is obtained in the form of c 2 Q n+1-Y for the number of algebraic numbers of degree n ≥ 3 and of height at most c 3 Q .
Highlights
Хаусдорфа [3] и доказательство аналога теоремы Хинчина в случае расходимости [4]
In this article, we study the distribution of points with real conjugate algebraic coordinates α = α1, α 2, α3 in V
Г. Проблема Малера в метрической теории чисел / В
Summary
Pn (Q) = {P(x) ∈ Z : deg P = n ≥ 2, H (P) ≤ Q}. В (1) степень полинома P(x=) an x n + an-1x n-1 + ... + a1x + a0 равна n, а высот= а H H= (P) max ai 0≤i≤n не превосходит Q. Класс Pn (Q) содержит (2Q +1) n+1 полиномов P(x). Задача состоит в поисках оценок количества векторов α, таких что α1, α 2 – корни P(x) ∈ Pn (Q), удовлетворяющих неравенству f (α1) - α 2 < Q -γ1. Которая в [9] была усилена до 0 ≤ γ1 < 3 / 4, а в [10] получено асимптотическое равенство. #L f (Q, γ1) >> Q n+1-γ1 , 0 ≤ γ1 < 1
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