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The theory of asymptotic distribution modulo one. (English) Zbl 0131.29202

MSC:
11K06 General theory of distribution modulo \(1\)
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11J71 Distribution modulo one
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[1] For references till 1986 cf. my Diophantische Approximationen (Berlin 1936, Ergebnisse der Mathematik IV, 4), in the following denoted by D. A. · JFM 62.0173.01
[2] J. Cigler und G. Helmberg , Neuere Entwicklungen der Theorie der Gleichverteilung . Jahresbericht der D.M.V. 64, 1-50 (1961). · Zbl 0109.03404
[3] A large part of: J.W.S. Cassels , An introduction to diophantine approximation (Cambridge Univ. Tract 45, 1957) also is dedicated to our subject. · Zbl 0077.04801
[4] E.g. P. Erdös in his contribution to this symposium: Problems and results on diophantine approximations , (this volume p. 52). |
[5] For this and similar formulae cf. my notes: Een algemeene stelling uit de theorie der gelijkmatige verdeeling modulo 1 . Mathematica (Zutphen) IIB, 7-11 (1942/43).
[6] Eenige integralen in de theorie der gelijkmatige verdeeling modulo 1 . Mathematica (Zutphen) 11B, 49-52 (1942/48).
[7] S. Bundgaard , Ueber de Werteverteilung der Charaktere abelscher Gruppen , Math.-fys. Medd. Danske Vid. Selsk. 14, No. 4, 1- 29 (1936). The author bases his work on VON NEUMANN’S notion of the mean value of an almost periodic function in a group (Transactions Amer. Math. Soc. 86, 445 - 492 (1934)). · Zbl 0015.00602
[8] B. Eckmann , Über monothetische Gruppen . Comment. math. Helvet. 16, 249- 268 (1948/44). · Zbl 0061.04402
[9] E.g. by L. Kuipers and B. Meulenbeld. For references cf. CIGLER-HELMBERG quoted in [2].
[10] For references cf. I.S. Gál - J.F. Koksma , Sur l’ordre de grandeur des fonctions sommables , Proc. Kon. Ned. Akad. Wet. 58, 638-653 (1950)= Indagationes Mathematicae 12, 192-207 (1950). · Zbl 0041.02406
[11] E.g. cf. P. Erdös - I.S. Gál , On the law of the iterated logarithm , Proc. Kon. Ned. Akad. Wet. 58, 65 - 84 (1955)= Indagationes Mathematicae 17, 65-84 (1955). · Zbl 0068.05403
[12] In his paper: Über die Gleichverteilung von Zahlen modulo Eins , Math. Ann. 77, 313 - 352 (1916) p. 845. · JFM 46.0278.06
[13] Cf. my paper: Asymptotische verdeling van reële getallen modulo 1 I, II, III , Mathematica, (Leiden) 1 (1988), 245 - 248,2 (1938), 1-6,8 (1933), 107-114 and D. A. Ch. VIII. · JFM 59.0958.02
[14] Part I (Zur Gleichverteilung modulo Eins) and Part II (Rhythmische Systeme, A und B) appeared in the Acta Math: J.G. Van Der Corput , Diophantische Ungleichungen , Acta Math. 56, 373-456 (1931),resp. 59, 209 - 328 (1932).
[15] K. Mahler , On the fractional parts of the powers of a rational number, I , Acta Arithm, 8 (1988), 89 - 93, II , Mathematika (London) 4 (1957), 122 -124.For further references concerning (26) etc. cf. the paper of PISOT-SALEM in this volume (p. 164). · Zbl 0208.31002
[16] I. Schoenberg , Ueber die asymptotische Verteilung reeller Zahlen mod. 1 . Math. Z. 28, 171-199 (1928). · JFM 54.0212.02
[17] R.J. Duffin and A.C. Schaeffer , Khintchine’s problems in metric Diophantine approximation . Duke Math. J. 8, 248-255 (1941). · Zbl 0025.11002
[18] J.F. Koksma , Niet-lineaire simultane approximaties . Handel. Ned. Nat. Congres, 95 - 96 (1941).
[19] Ibid. Sur la theorie métrique des approximations diophantiques , Proc. Ned. Akad. Wet. 48, 249 - 265 (1945).Indagationes Mathematicae 7, 54 - 70 (1945), where also further references are given. · Zbl 0060.12206
[20] J.W.S. Cassels , Some metrical theorems in diophantine approximation. I Proc. Cambr. Phil. Soc. 46, 209 - 218 (1949). II J. London Math. Soc. 25, 180 -184 (1950). · Zbl 0037.17201
[21] D. De Vries , Metrische onderzoekingen van Diophantische benaderingsproblemen in het niet-lacunaire geval . (Diss. Amsterdam, V.U.), 1955.
[22] J.G. Van Der Corput , Verteilungsfunktionen . Proc. Kon. Ned. Akad. Wet. 38, 813-821; 1058 -1060 (1988);89, 10-19; 19 - 26; 149-153; 339-344; 489- 494; 579 - 590 (1939).
[23] For references cf. K. Roth , On irregularities of distribution . Mathematika (London) 1, 73-79 (1954). · Zbl 0057.28604
[24] H. Davenport , Note on irregularities of distribution . Mathematika (London), 3, 131-135 (1956). · Zbl 0073.03402
[25] M. Tsuji , On the uniform distribution of numbers (mod. 1) . J. Math. Soc. Japan 4, 313-322 (1952). · Zbl 0048.03302
[26] For references cf. Dr. Cigler’S third paper in this vol. (p. 44). |
[27] N.M. Koroboff , Einige Probleme der Verteilung von Bruchteilen . Uspechi mat. Nauk 4, 189 -190 (1949).
[28] W. Leveque , On uniform distribution modulo a subdivision . Pacific J. of Math. 8, 757-771 (1953). · Zbl 0051.28503
[29] In this respect I mention a result by C. Ryll Nardzewski , Sur les suites et les fonctions également réparties . Studia math. 12,143 -144 (1951) which in certain cases gives a link between both theories. · Zbl 0042.28803
[30] It is the theorem which in its one dimensional case is quoted as Satz 4 in D.A. p. 101 and which itself is related to the old theorem of VAN DER CORPUT, which is meant in § 5a after (38) in this paper.For further references cf also [31]. Several applications a.o. are given by A. Drewes , Diophantische Benaderingsproblemen . (Diss. Amsterdam V.U.), 1945.
[31] P. Erdös and A. Turán , On a problem in the theory of uniform distribution I, II . Proc. Kon. Ned. Akad. Wet. (ser. A.) 51, 370-378; 406-413 (1948),= Indagationes Mathematicae 10, 370-378; 406 - 413 (1948). · Zbl 0031.25402
[32] J.F. Koksma , Some theorems on Diophantine inequalities . Scriptum 5 of the Mathematical Centre, Amsterdam (1950). · Zbl 0038.02803
[33] Cf. D. A. Ch. VIII, IX.
[34] J.W.S. Cassels , A new inequality with application to the theory of diophantine approximation . Math. Ann. 126, 108 -118 (1953). · Zbl 0051.28604
[35] Cf. D. A. IX, § 6, p. 116.
[36] Similar problems for generalized dyadic fractions have been treated by C. Sanders , Verdelingsproblemen bij gegeneraliseerde duale breuken . (Diss. Amsterdam V.U.), 1950.
[37] A. Khintchine , Asymptotische Gesetze der Wahrscheinlichkeitsrechnung , Ergebnisse der Mathematik II, 4, (1933). · JFM 59.1153.01
[38] Cf. [36] and e.g. W. Feller , An introduction to probability theory and its applications I, sec. ed. New York-London (1960). · Zbl 0138.10207
[39] P. Erdös and I.S. Gál , On the law of the iterated logarithm I, II . Proc. Kon. Ned. Akad. Wet. (ser. A), 58, 64-84 (1955),Indagationes Mathematicae 17, 64-84 (1955). · Zbl 0068.05403
[40] For ref. cf. e.g. my paper An arithmetical property of some sommable functions . Proc. Kon. Ned. Akad. Wet. (ser. A) 53, 960-972 (1950)= Indagationes Mathematicae 12, 354-367 (1950). · Zbl 0038.19102
[41] A. Khintchine , Eine arithmetische Eigenschaft der summierbaren Funktionen . Recueil Math., Moscou 41, 11-13 (1934). · Zbl 0009.30602
[42] C. Ryll-Nardzewski , On the ergodic theorems, I, II . Studia Mathematica XII, 65-79 (1951). · Zbl 0044.12302
[43] A proof of the first counter example in J.F. Koksma - R. Salem , Uniform distribution and Lebesgue integration . Acta Scient. Math. Szeged 12, 87-96 (1950). · Zbl 0036.03101
[44] A proof of the second counter example in P. Erdös , On the strong law of large numbers . Transactions Amer. Math. Soc. 67, 51- 56 (1950). · Zbl 0034.07201
[45] Cf. my paper: Sur les suites (\lambda n x) et les fonctions g(t) \in L(2) . J. de Math. p. appl. 85, 289 - 296 (1956). · Zbl 0070.28402
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