Trigonometric sums in the metric theory of Diophantine approximation
https://doi.org/10.22405/2226-8383-2019-20-2-207-220
Abstract
It is a survey with respect to using trigonometric sums in the metric theory of Diophantine approximation on the manifolds in n-dimensional Euclidean space. We represent both classical results and contemporary theorems for Γ, dim Γ = m,
n/2 < m < n. We also discuss reduction of a problem about Diophantine approximation to trigonometric sum or trigonometric integral, and indicate measure-theoretic considerations.
If m ≤ n/2 then usually it is used the other methods. For example, the essential and inessential domains method or methods of Ergodic Theory.
Here we cite two fundamental theorems of this theory. One of them was obtained by V. G. Sprindzuk (1977). The other theorem was proved by D. Y. Kleinbock and G. A. Margulis (1998). The first result was obtained using method of trigonometric sums. The second theorem was proved using methods of Ergodic Theory. Here the authors applied new technique which linked Diophantine approximation and homogeneous dynamics.
In conclusion, we add a short comment concerning the tendencies of a development of the metric theory of Diophantine approximation of dependent quantities and its contemporary aspects.
References
1. Bernik V. I., Kovalevskaja E. I. Extremal properties of some surface in n-dimensional Euclidean space, Math. Notes, 15(2). 1974, pp. 247–254.
2. Bernik V. I., Melnichuk Yu. V. Diophantine approximation and Hausdorff dimension. Nauka I Nechnika, Minsk, 1988. 23. Cassels J. W. S. An introduction to Diophantine Approximation. Cambridge Tracts in Math and Math. Phys., 45. Cambridge Univ. Press. 1957.
3. Cassels J. W. S. An introduction to Diophantine Approximation. Cambridge Tracts in Math and Math. Phys., 45. Cambridge Univ. Press. 1957.
4. Kovalevskaja E. I. "Hyperbolic" approximation on analytic manifolds, Dokl. Akad. Nauk BSSR, vol. 19(3), 1975, pp. 200–203.
5. Kovalevskaja E. I. Diophantine approximation with quadratic polynomials, Vesci Akad. Navuk BSSR. Ser. Fiz.-Mat. Navuk, 4, 1975, pp. 5–14.
6. Kovalevskaja E. I. One geometric property of extremal surface, Math. Notes, 23(2), 1978, pp. 177–181.
7. Kovalevskaya, E. I. The trigonometric sums and the metric theory of Diophantine approximation on manifolds, Proc. XV Int. Conf. on Algebra, Number Theory and Discrete Geometry: Contemporary Problems and Applications devoted to centenary of professor N. M. Korobov. Tula, Russia, 28–31 May 2018, P. 257–260. (In Russian)
8. Kavaleuskaya, E. I. Geometric and arithmetic description of extremal manifolds in the metric theory of Diophantine approximation, Proc. XVI Int. Conf. on Algebra, Number Theory and Discrete Geometry: Contemporary Problems, Applications and Problem of History, devoted to eighty of professor Mishel Deza. Tula, Russia, 13–18 May 2019, P. 239–241.
9. Kovalevskaya, E. I., Rykova O. V. The development of the essential and inessential domains method for the calculation of vectors with real algebraic coordinates near smooth surfaces, Chebyshevskii Sbornik. — Tula: 14, 2013, pp. 119–126.
10. Kubilius I. P. On the application of I. M. Vinogradov’s method to the solution of a problem of the metric theory of numbers, Dokl. Akad. Nauk SSSR, 67, 1949, pp. 783–786.
11. Kubilius I. P. On the metrical problem in the theory of Diophantine approximation, Trudy Akad. Nauk Litov. SSR, Ser. B, 2(18), 1959, pp. 3–7.
12. Sprindžuk V. G. Mahler’s problem in metric number theory. Minsk: Izdat. Nauka i Tehnika. 1967. English translation by B. Volkman. Transl. Math. Monographs, 25, Amer. Math. Soc., Providence, RI. 1969.
13. Sprindžuk V. G. The method of trigonometric sums in the metric theory of Diophantine approximation of dependent quantities, Proc. Steklov Inst. Math. Akad. Nauk SSSR, 128(2), 1972, pp. 212–228.
14. Sprindžuk V. G. Metric theory of Diophantine approximations. Izdat. Nauka. Moscow, 1977. English translation by R. A. Silverman. John Wiley and Sons. New York – Toronto – London, 1979.
15. Adiceam F., Beresnevich V., Levesley V., Velani S., Zorin E. Diophantine approximation and applications in interference alignment, Advances in Math., 302 2016, pp. 231–279.
16. Bayramoglu M., Jabbarov I. Sh., Kazimova L. G. On some theoretic-functional results concerning the theory of extremality and their application, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb., 44(2), 2018, pp. 229–237.
17. Beresnevich V., Bernik V., G‥otze F. Integral polynomials with small discriminants and resultants, Advances in Math., 298, 2016, pp. 393–412.
18. Beresnevich V., Ramirez F., Velani S. Metric Diophantine approximation: aspects on recent work. In Dynamics and Analytic Number Theory. LMS Lecture Notes Ser., 437, 2016 (eds. D. Badziahin, A. Gorodnik, N. Reyerimhoff). Cambridge Univ. Press (Cambridge. 2016), pp. 1–95.
19. Beresnevich V., Lee L., Vaughan R. C., Velani S. Diophantine approximation on manifolds and lower bounds Hausdorff dimension Math., 63, 2017, pp. 762–779.
20. Beresnevich V., Velani S. A note on three problems in metric Diophantine approximation. In recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., 631, 2015, Amer. Math. Soc. Providence. R. I., 2015, pp. 211–229.
21. Beresnevich V., Vaughan R. C., Velani S., Zorin E. Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory. Int. Math. Research Notices, 2016, pp. 1–24.
22. Bernik V. I., Dodson M. M. Metric Diophantine Approximation of Manifolds. Cambridge Tracts in Math, 137. Cambridge Univ. Press. Cambridge, 1999.
23. Bugeaud Y. Approximation by algebraic numbers. Cambridge Tracts in Math., 169. Cambridge Univ. Press, 2004.
24. Dodson M. M., Vickers J. A. G. Number Theory and dynamical systems, London Math. Soc. Lecture Note Ser., 134. Cambridge Univ. Press, 1989.
25. Kleinbock D. Y., Margulis G.A. Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math., 148, 1998, pp. 339–360.
26. Kovalevskaja E. I. Metric theorems on the approximation of zero by a linear combination of polynomials with integral coefficients, Acta Arith., 25, 1973, pp. 93–104.
27. Kovalevskaya E. The convergence part of a Khintchne-type theorem in the ring of adeles, Tatra Mountains Math. Publ., 59, 2014, pp. 39–50.
28. Schmidt W. M. Über Gitterpunkte auf gewissen Flächen, Monatch. Math. 68(1), 1964, pp. 59–74.
29. Schmidt W. M. Metrische Sätze Über simultane Approximationabhängiger Grössen, Monatch. Math., 68(2), 1964, pp. 154–166.
30. Schmidt W. M. Diophantine Approximation, Lecture Notes in Math., 785. Springer-Verlag, 1980.
31. Steuding J. Diophantine analysis. Course notes from a Summer School. Trends in Math., Birkhäuser. Springer Int. Publ. AG, 2016.
Review
For citations:
Kavaleuskaya E.I. Trigonometric sums in the metric theory of Diophantine approximation. Chebyshevskii Sbornik. 2019;20(2):207-220. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-2-207-220
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