This paper is a brief review of some general Diophantine results, best approximations and their applications to the theory of uniform distribution.
Cet article rassemble des résultats généraux d'approximation diophantienne, sur les meilleures approximations et leurs applications à la théorie de répartition uniforme.
@article{JTNB_1999__11_2_425_0, author = {Nikolai G. Moshchevitin}, title = {Continued fractions, multidimensional diophantine approximations and applications}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {425--438}, publisher = {Universit\'e Bordeaux I}, volume = {11}, number = {2}, year = {1999}, zbl = {0987.11043}, mrnumber = {1745888}, language = {en}, url = {https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_425_0/} }
TY - JOUR AU - Nikolai G. Moshchevitin TI - Continued fractions, multidimensional diophantine approximations and applications JO - Journal de théorie des nombres de Bordeaux PY - 1999 SP - 425 EP - 438 VL - 11 IS - 2 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_425_0/ LA - en ID - JTNB_1999__11_2_425_0 ER -
%0 Journal Article %A Nikolai G. Moshchevitin %T Continued fractions, multidimensional diophantine approximations and applications %J Journal de théorie des nombres de Bordeaux %D 1999 %P 425-438 %V 11 %N 2 %I Université Bordeaux I %U https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_425_0/ %G en %F JTNB_1999__11_2_425_0
Nikolai G. Moshchevitin. Continued fractions, multidimensional diophantine approximations and applications. Journal de théorie des nombres de Bordeaux, Volume 11 (1999) no. 2, pp. 425-438. https://jtnb.centre-mersenne.org/item/JTNB_1999__11_2_425_0/
[1] Klein's polyhedra for two cubic forms considered by Davenport. Matemeticheskie Zametki 56 (1994), 9-27. | MR | Zbl
& ,[2] An introduction to Diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics 45. Cambridge University Press, New York, 1957, x+166 pp. | MR | Zbl
,[3] Simultaneous Diophantine Approximations II. Proc. London Math. Soc. (3) 5 (1955), 435-448. | MR | Zbl
,[4] Zaremba's conjecture and sums of the divisor functions. Math. Comp. 61 (1993), no. 203, 171-176. | MR | Zbl
,[5] The Markoff and Lagrange spectra. AMS, Providence, Math. surveys and monographs 30, 1989. | MR | Zbl
, ,[6] Simultaneous Diophantine Approximations. Mathematika 1 (1954), 51-72. | MR | Zbl
,[7] An algorithm for the "divided cells" of a lattice. (Russian) Izvestiya Akad. Nauk SSSR. Ser. Mat. 11, (1947), 505-538. | MR | Zbl
,[8] The St. Petersburg school in the theory of numbers. (Russian) Akad. Nauk SSSR Naucno-Populamaya Seriya. Izdat Akad. Nauk SSSR, Moscow-Leningrad, 1947, 421 pp. | MR
,[9] Lattice Points. Pitman Monographs and Surveys in Pure and Applied Mathematics, 39. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. viii+184 pp. | MR | Zbl
, & ,[10] Geometry of Numbers. Amsterdam, 2nd edition, 1987. | MR | Zbl
& ,[11] Application of Number Theory to Numerical Analysis. Berlin; Heidelberg; N.Y.: Springer Verlag, 1981. | MR | Zbl
& ,[12] Uber die simultanen diophantischen Approximationen. Math. Zeitschr. 33 (1933), 505-543. | JFM | Zbl
,[13] Un théorème d'existence pour les approximations diophantiennes. Enseign. Math. (2) 15 (1969), 171-175. | MR | Zbl
,[14] Continued fractions. Moscow, 1978. | JFM
,[15] Über eine Klasse linearer Diophantischer Approximationen. Rendiconti Circ. Mat. Palermo 50 (1926), 170-195. | JFM
,[16] On the recurrence of an integral of an odd conditionally periodic function. (Russian) Mat. Zametki 61 (1997), no. 4, 570-577; translation in Math. Notes 61 (1997), no. 3-4, 473-479. | MR | Zbl
,[17] Representation of rationals by finite continuous fractions. (Russian) Uspekhi Mat. Nauk 51 (1996), no. 4(310), 159-160. | MR | Zbl
& ,[18] The integrals of quasiperiodic functions. (Russian) Vestnik Moskov. Univ. Ser. I Mat. Meh. 1978, no. 1, 106-115. | MR | Zbl
,[19] Methods of qualitative analysis in solid body motion. Moscow, Moscow University, 1980. | MR
,[20] Asymptotics of mean time values. Recent problems in theory of numbers and applications. Conference in Tula (Russia) 09 - 14.09.1996., Thesis., S. 80.
,[21] Two-dimensional continued fractions. The simplest ezamples. (Russian) Trudy Mat. Inst. Steklov. 209 (1995), 143-166. | MR | Zbl
,[22] Methods of theory of numbers in approximate analysis. Moscow, 1963.
,[23] On some problems in theory of Diophantine approximations. (Russian) Uspekhi Mat. Nauk 22 (1967), 83-118. | Zbl
,[24] Trigonometric sums and their applications. Moscow, 1989. | MR | Zbl
,[25] Uniformly distributed sequences. N.Y., 1974.
& ,[26] Best simultaneous Diophantine approximation II. Pac. J. Math. 102 (1982), 61-88. | MR | Zbl
,[27] Polyèdre d'Arnold et voile d'un cône simplicial: analogues du théorème de Lagrange. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 8, 711-716. | MR | Zbl
,[28] Généralisation de la théorie des fractions continues. Ann. Ecole norm. (3) 13 (1896), 41-60. | JFM | Numdam | MR
,[29] Distribution of values of linear functions and asymptotic behaviour of trajectories of some dynamical systems. (Russian) Matematicheskie Zametki 58 (1995), 394-410. | MR | Zbl
,[30] On the recurrence of an integral of a smooth three-frequency conditionally periodic function. (Russian) Mat. Zametki 58 (1995), no. 5, 723-735. | MR | Zbl
,[31] Nonrecurrence of an integral of a conditionally periodic function. (Russian) Mat. Zametki 49 (1991), no. 6, 138-140. | MR | Zbl
,[32] On representation of rationals as continuous fractions and sets free from arithmetic progressions. (Russian) Matematicheskie Zapiski 2 (1996), 99-109.
,[33] On simultaneous Diophantine approximations. Matemticheskie Zametki 61 (1997), no. 5, 706-716. | MR | Zbl
,[34] On the recurrence of an integral of a smooth conditionally periodic function. (Russian) Mat. Zametki 63 (1998), no. 5, 737-748. | MR | Zbl
,[35] On geometry of the best approximations. Doklady Rossiiskoi Akademii Nauk 359 (1998), 587-589. | MR | Zbl
,[36] On best joint approximations. (Russian) Uspekhi Mat. Nauk 51 (1996), no. 6(312), 213-214. | MR | Zbl
,[37] Recent results on asymptotic behaviour of integrals of quasiperiodic functions. Amer. Math. Soc. Transl. (2) 168 (1995), 201-209. | MR | Zbl
,[38] Application of Diophantine Approximation to Numerical Integration. in book Dioph. Appr. and Appl. ed. Osgood C. N.Y., Academic Press, 1973, 129 -199. | MR | Zbl
,[39] Dyadic Fractions with Small Partial Quotients. Monatsh. Math. 101 (1986), 309-315. | MR | Zbl
,[40] On Uniform Distribution of Algebraic Numbers. Proc. Amer. Math. Soc. 4 (1953), 440-443. | MR | Zbl
,[41] Sur les séries trigonométriques. Comptes Rendus 101 (1885), 1131-1134. | JFM
,[42] On curves defined by differential equations. (Russian transl.) Moscow-Leningrad, 1947.
[43] Diophantine approximation. Lecture Notes in Mathematics 785. Springer, Berlin, 1980. x+299 pp. | MR | Zbl
,[44] On conditions of uniform Poisson stability of cylindrical systems. Uspekhi Mat. Nauk 34 (1979), no. 6(210), 184-188. | MR | Zbl
,[45] Minima of decomposable cubic forms in 3 variables. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 168 (1988), 125-139. | MR | Zbl
,[46] Minima of decomposable forms of degree n in n variables. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 183 (1990), 142-154. | MR | Zbl
,[47] Metric theory of diophantine approximations. Moscow, 1977. | Zbl
,[48] Asymptotic behavior of integrals of quasiperiodic functions. (Russian) Differential Equations 3 (1967), 862-868. | MR | Zbl
,[49] Estimates of the remainder term in Kronecker theorem on approximations. Theoriia Functzii, Funktzianal. Analiz i ih prilogeniia, Harkov, 5 (1967), 84-98.
,[50] On one generalization of continued fractions algorithm. Selected works T.1, Kiev, 1952.
,[51] Über die Gleichverteliung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313-352. | JFM | MR
,[52] La méthode des "bons treillis" pour le calcul des intégrales multiples. In: Appl. of Number Theory to Numerical Analysis. ed. S.K. Zaremba; N.Y., 1972, 39-119. | MR | Zbl
,