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BADLY APPROXIMABLE MATRICES AND DIOPHANTINE EXPONENTS

https://doi.org/10.22405/2226-8383-2013-14-4-38-79

Abstract

This paper is a survey of results concerning different kinds of Diophantine exponents. Special attention is paid to the transference principle and to generalization of the concept of badly approximable numbers to matrices and lattices.

 

About the Author

O. N. German
Московский государственный университет имени М. В. Ломоносова
Russian Federation


References

1. Apfelbeck A. A contribution to Khintchine’s principle of transfer // Czech. Math. J. 1951. Vol. 1, № 3. P. 119–147.

2. Arnold V. I. Higher dimensional continued fractions // Regular and Chaotic Dynamics. 1998. Vol. 3, № 3.

3. Arnold V. I. Preface // Amer. Math. Soc. Transl. 1999. Vol. 197, № 2. P. ix–xii.

4. Arnold V. I. Continued fractions. Moscow: Moscow Center of Continuous Mathematical Education, 2002.

5. Ball K. Volumes of sections of cubes and related problems // Geometric aspects of functional analysis (1987–88). Lect. Notes in Math. 1989. Vol. 1376. P. 251– 260.

6. Bonnesen T., FenchelW. Theorie der konvexen K¨orper. Berlin: Springer, 1934.

7. Боревич З. И., Шафаревич И. Р. Теория чисел. М.: Наука, 1964. 566 с.

8. Bugeaud Y., Laurent M. On transfer inequalities in Diophantine approximations II // Math. Z. 2010. Vol. 265, № 2. P. 249–262.

9. Cassels J.W. S. An introduction to Diophantine approximation. Cambridge University Press, 1957.

10. Cassels J.W. S., Swinnerton-Dyer H. P. F. On the product of three homogeneous linear forms and indefinite ternary quadratic forms // Philos. Trans. Roy. Soc. London. Ser. A. 1955. Vol. 248. P. 73–96.

11. Dyson F. J. On simultaneous Diophantine approximations // Proc. London Math. Soc. 1947. Vol. 49, № 2. P. 409–420.

12. Ewald G. Combinatorial convexity and algebraic geometry. Springer, 1996. 13. Герман О. Н. Паруса и норменные минимумы решеток // Мат. Сборник. 2005. Т. 196, №3. С. 31–60.

13. Герман О. Н. Полиэдры Клейна и норменные минимумы решеток // Доклады РАН. Математика. 2006. Т. 406, №3. С. 38–41.

14. German O. N. Klein polyhedra and lattices with positive norm minima // Jourbal de Th´eorie des Nombres de Bordeaux. 2007. Vol. 19. P. 157–190.

15. Герман О. Н., Лакштанов Е. Л. О многомерном обобщении теоремы Лагранжа для цепных дробей // Известия РАН. Сер. Мат. 2008. Т. 72, №1. С. 51–66.

16. German O. N. Transference inequalities for multiplicative Diophantine exponents // Труды МИРАН. 2011. Т. 275. С. 216–228.

17. German O. N. On Diophantine exponents and Khintchine’s transference principle // Moscow Journal of Combinatorics and Number Theory. 2012. Vol. 2, № 2. P. 22–51.

18. German O. N. Intermediate Diophantine exponents and parametric geometry of numbers // Acta Arithmetica. 2012. Vol. 154. P. 79–101.

19. Gr¨unbaum B. Convex polytopes. London; New York; Sydney: Interscience Publ., 1967.

20. Jarn´ık V. Uber einen Satz von A. Khintchine // Prace Mat. Fiz. 1936. Vol. 43. ¨ P. 151–166.

21. Jarn´ık V. Uber einen Satz von A. Khintchine, 2 // Acta Arithm. 1936. Vol. 2. ¨ P. 1–22.

22. Jarn´ık V. Zum Khintchineschen “Ubertragungssatz” // Trav. Inst. Math. ¨ Tbilissi. 1938. Vol. 3. P. 193–212.

23. Khintchine A. Ya. Uber eine Klasse linearer Diophantischer Approximationen. ¨ Rend. Sirc. Mat. Palermo. 50 (1926). P. 170–195.

24. Khintchine A. Ya. On some applications of the method of the additional variable // UMN. 1948. Vol. 3, № 6(28). P. 188–200.

25. Klein F. Uber eine geometrische Auffassung der gewohnlichen Kettenbruchentwichlung // Nachr. Ges. Wiss. Gottingen. 1895. Vol. 3. P. 357–359.

26. Korkina E. La p´eriodecit´e des fractions continues multidimensionnelles // C. R. Acad. Sci. Paris. S´erie I. 1994. Vol. 319. P. 777–780.

27. Korkina E. I. Two-dimensional continued fractions. The simplest examples // Proc. Steklov Math. Inst. RAS. 1995. Vol. 209. P. 143–166.

28. Lachaud G. Poly`edre d’Arnol’d et voile d’un cˆone simplicial: analogues du th´eor`eme de Lagrange // C. R. Acad. Sci. Paris. 1993. S´erie I. Vol. 317. P. 711–716.

29. Lachaud G. Sails and Klein Polyhedra // Contemporary Mathematics. 1998. Vol. 210. P. 373–385.

30. Lachaud G. Voiles et Poly`edres de Klein. Act. Sci. Ind. Hermann, 2002.

31. Laurent M. On transfer inequalities in Diophantine Approximation // “Analytic Number Theory, Essays in Honour of Klaus Roth” / eds. W.W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt and R. C. Vaughan. Cambridge University Press, 2009. P. 306–314.

32. Laurent M. Exponents of Diophantine approximation in dimension two // Canad. J. Math. 2009. Vol. 61. P. 165–189.

33. McMullen P., Shephard G. C. Convex polytopes and the upper bound conjecture. Cambridge (GB): Cambridge University Press, 1971.

34. Mahler K. Ein Ubertragungsprinzip f¨ur lineare Ungleichungen // ¨ Cas. Peˇst. Mat. ˇ Fys. 1939. Vol. 68. P. 85–92.

35. Mahler K. On a theorem of Dyson // Mat. sbornik. 1950. Vol. 26(68), № 3. P. 457–462.

36. Mahler K. On compound convex bodies (I). // Proc. London Math. Soc. 1955. Vol. 5, № 3. P. 358–379.

37. Moshchevitin N. G. Khintchine’s singular systems and their applications // UMN (to appear). Preprint available at arXiv:0912.4503v1, 2009.

38. Moussafir J.-O. Convex hulls of integral points // Zapiski nauch. sem. POMI. 2000. Vol. 256.

39. SchmidtW. M. On heights of algebraic subspaces and diophantine approximations // Annals of Math. 1967. Vol. 85, № 3. P. 430–472.

40. SchmidtW. M. Diophantine Approximation. Lecture Notes in Math. Vol. 785. Springer-Verlag, 1980.

41. SchmidtW. M., Summerer L. Parametric geometry of numbers and applications // Acta Arithmetica. 2009. Vol. 140, № 1. P. 67–91.

42. SchmidtW. M., Summerer L. Diophantine approximation and parametric geometry of numbers // Monat. Math. 2012. DOI: 10.1007/s00605-012-0391-z.

43. SchmidtW. M., Wang Y. A note on a transference theorem of linear forms // Sci. Sinica. 1979. Vol. 22, № 3. P. 276–280.

44. Скубенко Б. Ф. Минимумы разложимой кубической формы от трех переменных // Зап. науч. семинара ЛОМИ. Т. 168. 1988.

45. Скубенко Б. Ф. Минимумы разложимых форм степени n от n переменных при n > 3 // Зап. науч. семинара ЛОМИ. Т. 183. 1990.

46. Tsuchinashi H. Higher dimensional analogues of periodic continued fractions and cusp singularities // Tohoku Math. Journal. 1983. Vol. 35. P. 607–639.

47. Vaaler J. D. A geometric inequality with applications to linear forms // Pacif. J. Math. 1979. Vol. 83, №2. P. 543–553.

48. Waldschmidt M. Report on some recent advances in Diophantine approximation. Special volume in honor of Serge Lang. Springer Verlag, (to appear). Preprint available at arXiv:0908.3973v1. 2009.

49. Wang Y., Yu K. A note on some metrical theorems in Diophantine approximation // Chinese Ann. Math. 1981. Vol. 2. P. 1–12.


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German O.N. BADLY APPROXIMABLE MATRICES AND DIOPHANTINE EXPONENTS. Chebyshevskii Sbornik. 2013;14(4):38-79. (In Russ.) https://doi.org/10.22405/2226-8383-2013-14-4-38-79

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