Complex algebraic numbers in the sets of $\mathbb{C}^2$ of small Lebesgue measure
Trudy Instituta matematiki, Tome 26 (2018) no. 1, pp. 25-30.

Voir la notice de l'article provenant de la source Math-Net.Ru

Algebraic numbers of degree $n$ are investigated. For any $Q \ge {Q_0}\left( n \right)$ we show lower bound for distribution of complex algebraic numbers of height less then $Q$ near a smooth curve $f(z)$. We prove that for a set of points satisfying the condition $|f(\alpha _{1})- \alpha _{2}|$ their quantity is bounded below by $c_{15}Q^{n+1- \gamma }$.
@article{TIMB_2018_26_1_a4,
     author = {V. I. Bernik and M. A. Zhur},
     title = {Complex algebraic numbers in the sets of $\mathbb{C}^2$ of small {Lebesgue} measure},
     journal = {Trudy Instituta matematiki},
     pages = {25--30},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2018_26_1_a4/}
}
TY  - JOUR
AU  - V. I. Bernik
AU  - M. A. Zhur
TI  - Complex algebraic numbers in the sets of $\mathbb{C}^2$ of small Lebesgue measure
JO  - Trudy Instituta matematiki
PY  - 2018
SP  - 25
EP  - 30
VL  - 26
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TIMB_2018_26_1_a4/
LA  - ru
ID  - TIMB_2018_26_1_a4
ER  - 
%0 Journal Article
%A V. I. Bernik
%A M. A. Zhur
%T Complex algebraic numbers in the sets of $\mathbb{C}^2$ of small Lebesgue measure
%J Trudy Instituta matematiki
%D 2018
%P 25-30
%V 26
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TIMB_2018_26_1_a4/
%G ru
%F TIMB_2018_26_1_a4
V. I. Bernik; M. A. Zhur. Complex algebraic numbers in the sets of $\mathbb{C}^2$ of small Lebesgue measure. Trudy Instituta matematiki, Tome 26 (2018) no. 1, pp. 25-30. http://geodesic.mathdoc.fr/item/TIMB_2018_26_1_a4/

[1] Gelfond A. O., Transtsendentnye i algebraicheskie chisla, GITTL, M., 1952 | MR

[2] Kassels Dzh.V.S., Vvedenie v teoriyu diofantovykh priblizhenii, IL, M., 1961, 10 pp.

[3] Schneider Th., Einfuehrung in die Transzendenten Zahlen, Springer-Verlag, 1957, 139 pp. | MR

[4] Mahler K., “Ueber das Mass der Menge aller S-Zahlen”, Math. Ann., 1932, 106 | MR | Zbl

[5] Sprindzhuk V. G., Problema Malera v metricheskoi torii chisel, Nauka i tekhnika, Minsk, 1967 | MR

[6] Sprindzhuk V. G., “Dokazatelstvo gipotezy Malera o mere mnozhestva S-chisel”, Izv. AN SSSR. Ser. matem., 29:2 (1965) | MR | Zbl

[7] Bernik V. I., “Metricheskaya teorema o sovmestnom priblizhenii nulya znacheniyami tselochislennykh mnogochlenov”, Izv. AN SSSR. Ser. matem., 44:1 (1980) | Zbl

[8] Bernik V. I., “Application of Hausdorff Dimension in the theory of Diophantine Approximation”, Acta Arithmetica, 42:3 (1983), 219–253 | DOI | MR | Zbl

[9] Beresnevich V., “On approximation of real numbers by real algebraic numbers”, Acta Arithmetica, 90 (1999), 97–112 | DOI | MR | Zbl

[10] Bernik V. I., Gettse F., “Raspredelenie deistvitelnykh algebraicheskikh chisel proizvolnoi stepeni v korotkikh intervalakh”, Izv. RAN. Ser. matem., 79:1 (2015), 21–42 | DOI | MR | Zbl

[11] Beresnevich V., Dickinson D., Velani S., “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. Of Math., 166:2 (2007), 367–426 | DOI | MR | Zbl

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

[13] Lamchanovskaya M. V., Kalosha N. I., “O raspredelenii kompleksnykh algebraicheskikh chisel v krugakh malogo radiusa na kompleksnoi ploskosti”, Tr. In-ta matem., 23:1 (2015), 85 | MR

[14] Bernik V., Gotze F., Gusakova A., “On points with algebraically conjugate coordinates close to smooth curves”, Moskow Journal of Combinatirics and Number Theory, 2:3 (2016), 56–101 | MR | Zbl