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.

Bibliographie

[1] Bernik V., Goetze F., Kukso O., “On algebraic points in the plane near smooth curves”, Lith. Math. Journal, 54:3 (2014), 231–251 | DOI | MR | Zbl

[2] Schmidt W., “T-numbers do exist”, Symposia Mathematica, IV (INDAM, Rome, 1968/1969), 1970, 3–26 | MR | Zbl

[3] Budarina N. V., Bernik V. I., O'Donnell H., “Real algebraic numbers of the third degree in a short intervals”, Doklady NAN Belarusi, 57:4 (2012), 23–26

[4] Beresnevich V., Bernik V., Goetze F., “The distribution of close conjugate algebraic numbers”, Compos. Math., 146:5 (2010), 1165–1179 | DOI | MR | Zbl

[5] Beresnevich V. V., Bernik V. I., Goetze F., “On distribution of the values of resultants of integer polynomials”, Doklady NAN Belarusi, 54:5 (2010), 21–23 | MR

[6] Beresnevich V. V., Bernik V. I., Goetze F., “Conjugate approximations of zero by integer polynomial, its derivative and small values of discriminants”, Doklady NAN Belarusi, 54:2 (2010), 26–27

[7] Beresnevich V., “Rational points ear manifolds and metric Diophantine approximation”, Ann. of Math., 175:1 (2012), 187–235 | DOI | MR | Zbl

[8] Bugeaud Y., Approximation by algebraic numbers, Cambridge Tracts in Mathematics, 160, Cambridge University Press, Cambridge, 2004, 274 pp. | MR | Zbl

[9] Feldman N. I., “Approximation of some transcendental numbers. I: Approximation of logarithms of algebraic numbers”, Izv. AN SSSR, 15:1 (1951), 53–74 | MR

[10] Sprindzhuk V. G., Mahlers's problem in metric number theory, Nauka i tehnika, Minsk, 1967, 184 pp. (in Russian)

[11] Bernik V. I., “Using Hausdorff dimension in the theory of Diophantine approximation”, Acta Arith., 42:3 (1983), 219–253 | MR | Zbl

[12] Beresnevich V. V., Vaughan R. C., Velani S. L., “Inhomogeneous Diophantine approximation on planar curves”, Math. Ann., 349:4 (2011), 929–942 | DOI | MR | Zbl

[13] Beresnevich V., Dickinson D., Velani S., “Diophantine approximation on planar curves and the distribution of rational points”, Ann. of Math., 166:2 (2007), 367–426 | DOI | MR | Zbl

[14] Bernik V., Beresnevich V., Goetze F., Kukso O., “Distribution of algebraic numbers and metric theory of Diophantine approximation”, Limit theorems in probability, statistics and number theory, Springer Proc. Math. Stat., 42, Springer, Heidelberg, 2013, 23–48 | MR | Zbl

[15] Beresnevich V., Zorin E., “Explizit bounds for rational points near planar curves and metric Diophantine approximation”, Adv. Math., 225:6 (2010), 3064–3087 | DOI | MR | Zbl

[16] Budarina N. V., “The Mahler problem with nonmonotone right-hand side in field of complex numbers”, Math. Notes, 93:5–6 (2013), 812–820 | MR | Zbl