×

On Diophantine approximations. (English) Zbl 0100.03904


Keywords:

number theory
Full Text: DOI

References:

[1] J. W. S. Cassels, Über \varliminf_{\?\to +\infty }\?|\?\?+\?-\?|, Math. Ann. 127 (1954), 288 – 304 (German). · Zbl 0055.04401 · doi:10.1007/BF01361127
[2] -, An introduction to Diophantine approximations, Cambridge Tracts No. 45, Cambridge, 1957.
[3] J. H. H. Chalk, Rational approximations in the complex plane. II, J. London Math. Soc. 31 (1956), 216 – 221. · Zbl 0072.03901 · doi:10.1112/jlms/s1-31.2.216
[4] Roger Descombes, Sur la répartition des sommets d’une ligne polygonale régulière non fermée, Ann. Sci. Ecole Norm. Sup. (3) 73 (1956), 283 – 355 (French). · Zbl 0072.03802
[5] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. · Zbl 0058.03301
[6] S. Hartman, Sur une condition supplémentaire dans les approximations diophantiques, Colloquium Math. 2 (1949), 48 – 51 (French). · Zbl 0038.18802
[7] Edmund Hlawka, Über die Approximation von zwei komplexen inhomogenen Linearformen, Monatsh. Math. Phys. 46 (1937), no. 1, 324 – 334 (German). · JFM 64.0146.02 · doi:10.1007/BF01792688
[8] A. Hurwitz, Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche, Math. Ann. 39 (1891), no. 2, 279 – 284 (German). · JFM 23.0222.02 · doi:10.1007/BF01206656
[9] A. Khintchine, Neuer Beweis und Verallgemeinerung eines Hurwitzschen Satzes, Math. Ann. 111 (1935), no. 1, 631 – 637 (German). · Zbl 0012.24705 · doi:10.1007/BF01472245
[10] J. F. Koksma, Sur l’approximation des nombres irrationnels sous une condition supplémentaire, Simon Stevin 28 (1951), 199 – 202 (French). · Zbl 0044.04303
[11] L. Kuipers and B. Meulenbeld, Some properties of continued fractions, Acta Math. 87 (1952), 1 – 12. · Zbl 0046.04604 · doi:10.1007/BF02392279
[12] W. J. LeVeque, On asymmetric approximations, Michigan Math. J. 2 (1954), 1 – 6. · Zbl 0059.03901
[13] Kurt Mahler, On the product of two complex linear polynomials in two variables, J. London Math. Soc. 15 (1940), 213 – 236. · Zbl 0027.15802 · doi:10.1112/jlms/s1-15.3.213
[14] Max Müller, Über die Approximation reeller Zahlen durch die Näherungsbrüche ihres regelmässigen Kettenbruches, Arch. Math. 6 (1955), 253 – 258 (German). · Zbl 0064.04401 · doi:10.1007/BF01899402
[15] Ivan Niven, Irrational numbers, The Carus Mathematical Monographs, No. 11, The Mathematical Association of America. Distributed by John Wiley and Sons, Inc., New York, N.Y., 1956. · Zbl 0070.27101
[16] Nicolae Negoescu, Quelques précisions concernant le théorème de M. B. Segre sur des approximations asymétriques des nombres irrationnels par les rationnels, Bull. École Polytech. Jassy [Bul. Politehn. Gh. Asachi. Iaşi] 3 (1948), 3 – 16 (French). · Zbl 0032.40101
[17] Nicolae Negoescu, Note on a theorem of unsymmetric approximation, Acad. Repub. Pop. Române. Bul. Sti. A. 1 (1949), 115 – 117 (Romanian).
[18] N. Obrechkoff, Sur l’approximation des nombres irrationnels par des nombres rationnels, C. R. Acad. Bulgare Sci. 3 (1950), no. 1, 1 – 4 (1951) (Russian, with French summary). · Zbl 0043.27702
[19] C. D. Olds, Note on an asymmetric Diophantine approximation, Bull. Amer. Math. Soc. 52 (1946), 261 – 263. · Zbl 0060.11809
[20] Alexander Oppenheim, Rational approximations to irrationals, Bull. Amer. Math. Soc. 47 (1941), 602 – 604. · Zbl 0027.16104
[21] A. V. Prasad, Note on a theorem of Hurwitz, J. London Math. Soc. 23 (1948), 169 – 171. · Zbl 0034.02803 · doi:10.1112/jlms/s1-23.3.169
[22] Raphael M. Robinson, The approximation of irrational numbers by fractions with odd or even terms, Duke Math. J. 7 (1940), 354 – 359. · Zbl 0024.25201
[23] Raphael M. Robinson, Unsymmetrical approximation of irrational numbers, Bull. Amer. Math. Soc. 53 (1947), 351 – 361. · Zbl 0032.40003
[24] W. T. Scott, Approximation to real irrationals by certain classes of rational fractions, Bull. Amer. Math. Soc. 46 (1940), 124 – 129. · Zbl 0022.30803
[25] B. Segre, Lattice points in infinite domains and asymmetric Diophantine approximations, Duke Math. J. 12 (1945), 337 – 365. · Zbl 0060.11807
[26] Vera T. Sós, On the theory of Diophantine approximations. II. Inhomogeneous problems, Acta Math. Acad. Sci. Hungar 9 (1958), 229 – 241. · Zbl 0086.03902 · doi:10.1007/BF02023874
[27] Leonard Tornheim, Asymmetric minima of quadratic forms and asymmetric Diophantine approximation, Duke Math. J. 22 (1955), 287 – 294. · Zbl 0064.28303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.