Geodesic
On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 14-47 
V. Bernik; F. Götze; A. Gusakova. On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXVII, Tome 448 (2016), pp. 14-47. http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a1/
@article{ZNSL_2016_448_a1,
     author = {V. Bernik and F. G\"otze and A. Gusakova},
     title = {On the distribution of points with algebraically conjugate coordinates in a~neighborhood of smooth curves},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {14--47},
     year = {2016},
     volume = {448},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a1/}
}
TY  - JOUR
AU  - V. Bernik
AU  - F. Götze
AU  - A. Gusakova
TI  - On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2016
SP  - 14
EP  - 47
VL  - 448
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a1/
LA  - en
ID  - ZNSL_2016_448_a1
ER  - 
%0 Journal Article
%A V. Bernik
%A F. Götze
%A A. Gusakova
%T On the distribution of points with algebraically conjugate coordinates in a neighborhood of smooth curves
%J Zapiski Nauchnykh Seminarov POMI
%D 2016
%P 14-47
%V 448
%U http://geodesic.mathdoc.fr/item/ZNSL_2016_448_a1/
%G en
%F ZNSL_2016_448_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $\varphi\colon\mathbb R\to\mathbb R$ be a continuously differentiable function on a finite interval $J\subset\mathbb R$, and let $\boldsymbol\alpha=(\alpha_1,\alpha_2)$ be a point with algebraically conjugate coordinates such that the minimal polynomial $P$ of $\alpha_1,\alpha_2$ is of degree $\leq n$ and height $\leq Q$. Denote by $M^n_\varphi(Q,\gamma,J)$ the set of points $\boldsymbol\alpha$ such that $|\varphi(\alpha_1)-\alpha_2|\leq c_1Q^{-\gamma}$. We show that for $0<\gamma<1$ and any sufficiently large $Q$ there exist positive values $c_2, where $c_i=c_i(n)$, $i=1,2$, that are independent of $Q$ and such that $c_2\cdot Q^{n+1-\gamma}<\# M^n_\varphi(Q,\gamma,J).

[1] V. Beresnevich, D. Dickinson, S. Velani, “Diophantine approximation on planar curves and the distribution of rational points”, (with an appendix “Sums of two squares near perfect squares” by R. C. Vaughan), Ann. Math., 166:2 (2007), 367–426 | DOI | MR | Zbl

[2] V. I. Bernik, “A metric theorem on the simultaneous approximation of zero by values of integer polynomials”, Izv. Akad. Nauk SSSR Ser. Mat., 44:1 (1980), 24–45 | MR | Zbl

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

[4] V. I. Bernik, F. Götze, “Distribution of real algebraic numbers of arbitrary degree in short intervals”, Izv. Math., 79:1 (2015), 18–39 | DOI | MR | Zbl

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

[6] Y. Bugeaud, Approximation by Algebraic Numbers, Cambridge Univ. Press, Cambridge, 2004 | MR | Zbl

[7] M. N. Huxley, Area, Lattice Points, and Exponential Sums, Oxford Univ. Press, New York, 1996 | MR | Zbl

[8] V. G. Sprindzuk, Mahler's Problem in Metric Number Theory, Amer. Math. Soc., Providence, RI, 1969 | MR

[9] W. M. Schmidt, Diophantine Approximation, Lect. Notes Math., 785, Springer, Berlin, 1980 | MR | Zbl

[10] N. I. Fel'dman, “The approximation of certain transcendental numbers. I. Approximation of logarithms of algebraic numbers”, Izv. Akad. Nauk SSSR, Ser. Mat., 15:1 (1951), 53–74 | MR | Zbl

[11] K. Mahler, “An inequality for the discriminant of a polynomial”, Michigan Math. J., 11 (1964), 257–262 | DOI | MR | Zbl

[12] J. F. Koksma, “Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen”, Monatsh. Math. Physik, 48 (1939), 176–189 | DOI | MR | Zbl

[13] B. L. van der Waerden, Algebra, Springer-Verlag, Berlin–Heidelberg, 1971 | Zbl

[14] R. C. Vaughan, S. Velani, “Diophantine approximation on planar curves: the convergence theory”, Invent. Math., 166:1 (2006), 103–124 | DOI | MR | Zbl

[15] V. I. Bernik, M. M. Dodson, Metric Diophantine Approximation on Manifolds, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[16] N. A. Pereverzeva, “The distribution of vectors with algebraic coordinates in $\mathbb R^2$”, Vestsi Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk, 1987, no. 4, 114–116, 128 | MR | Zbl

[17] V. Bernik, F. Götze, A. Gusakova, “On points with algebraically conjugate coordinates close to smooth curves”, Moscow J. Combin. Number Theory (to appear)