Distribution of algebraic points in domains of small measure and near the surfaces

V. I. Bernik ; A. G. Gusakova ; A. V. Ustinov

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 78-94

Résumé

Some questions about distribution of the points with rational coordinates are natural generalizations of problems about integer points in convex domains. Upper and lower bounds for the quantity of rational points on the circle were used in the study of Hausdorff dimension of the set of the point on circle which are approximated with a given order of accuracy by the points with rational coordinates. During the last 15 years in the papers of M. Huxley, V. I. Bernik, V. V. Beresnevich, S. Velani, R. Vaughan double sided asymptotic estimates for the quantity of rational points near the smooth curves and surfaces were found. Let $I=[a,b]\in\mathbb{R}$ is an interval, $y=f(x)$ is twice continuously differentiable function which satisfies the inequality $$ c_1|f''(x)|$$ for $c_2>c_1>0$ and for all $x\in I$. For arbitrary $\gamma$, $0\leq\gamma 1$ for sufficiently large $Q$ we denote by $A_I(Q,\gamma)$ the set of rational points $\Gamma=\left(\frac{p_1}{q},\frac{p_2}{q}\right)$, $aq\leq p_1\leq bq$, $1\leq q\leq Q$, for witch the following inequality holds $$ \left|f\left(\frac{p_1}{q}\right)-\frac{p_2}{q}\right|^{-1-\gamma}. $$ The set $A_I(Q,\gamma)$ consists from points lying inside the strip width of $2Q^{-\gamma}$ near the curve $y=f(x)$, $x\in I$. It it natural to expect that $\#A_I(Q,\gamma)$ is a value of the order $Q^{3-\gamma}$. It was proved using the methods of geometry of numbers and metric theory of Diophantine approximations. Recently [1] new estimates of the quantity of points $\bar{\alpha}=(\alpha_1,\alpha_2)\in\mathbb{R}^2$, where $\alpha_1, \alpha_2$ are conjugate real algebraic numbers of arbitrary degree $\deg\alpha_1=\deg\alpha_2=n$ and of the height $H(\alpha_1)=H(\alpha_2)\leq Q$, in the strip width of $c(n)Q^{-\gamma}$, $0\leq\gamma\leq\frac12$, $Q>Q_0(n)$ near the smooth curve $y=f(x)$ were obtained. In our paper some new results about distribution of points with conjugate real and complex algebraic coordinates were obtained. In particular generalization of result mentioned above was obtained. The main idea of the proof is based on metric theorem about diophantine approximations in the domains $G$ of small measure $\mu G$, $0\leq\gamma_1\leq\frac13$. Bibliography: 16 titles.

Détail

Keywords: Diophantine approximations, Lebesgue measure, conjugate algebraic numbers, height of algebraic number.