The theory of asymptotic distribution modulo one
Compositio Mathematica, Tome 16 (1964), pp. 1-22 
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     title = {The theory of asymptotic distribution modulo one},
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     pages = {1--22},
     year = {1964},
     publisher = {Kraus Reprint},
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     mrnumber = {173664},
     zbl = {0131.29202},
     language = {en},
     url = {https://www.numdam.org/item/CM_1964__16__1_0/}
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Koksma, J. F. The theory of asymptotic distribution modulo one. Compositio Mathematica, Tome 16 (1964), pp. 1-22. https://www.numdam.org/item/CM_1964__16__1_0/

[1] For references till 1986 cf. my Diophantische Approximationen (Berlin 1936, Ergebnisse der Mathematik IV, 4), in the following denoted by D. A. | JFM

[2] J. Cigler und G. Helmberg, Neuere Entwicklungen der Theorie der Gleichverteilung. Jahresbericht der D.M.V. 64, 1-50 (1961). | Zbl | MR

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 | MR

[3] E.g. P. Erdös in his contribution to this symposium: Problems and results on diophantine approximations, (this volume p. 52). | Numdam

[4] 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).

Eenige integralen in de theorie der gelijkmatige verdeeling modulo 1. Mathematica (Zutphen) 11B, 49-52 (1942/48).

[5] 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

B. Eckmann, Über monothetische Gruppen. Comment. math. Helvet. 16, 249- 268 (1948/44). | Zbl | MR

[8] E.g. by L. Kuipers and B. Meulenbeld. For references cf. CIGLER-HELMBERG quoted in [2].

[9] 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 | MR

[10] 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

[11] In his paper: Über die Gleichverteilung von Zahlen modulo Eins, Math. Ann. 77, 313 - 352 (1916) p. 845. | MR | JFM

[12] 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

[13] 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). | Zbl | JFM

[14] 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

[15] I. Schoenberg, Ueber die asymptotische Verteilung reeller Zahlen mod. 1. Math. Z. 28, 171-199 (1928). | MR | JFM

[18] R.J. Duffin and A.C. Schaeffer, Khintchine's problems in metric Diophantine approximation. Duke Math. J. 8, 248-255 (1941). | Zbl | JFM

J.F. Koksma, Niet-lineaire simultane approximaties. Handel. Ned. Nat. Congres, 95 - 96 (1941).

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 | MR

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 | MR

[19] D. De Vries, Metrische onderzoekingen van Diophantische benaderingsproblemen in het niet-lacunaire geval. (Diss. Amsterdam, V.U.), 1955.

[20] 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). | Zbl | JFM

[21] For references cf. K. Roth, On irregularities of distribution. Mathematika (London) 1, 73-79 (1954). | Zbl

[22] H. Davenport, Note on irregularities of distribution. Mathematika (London), 3, 131-135 (1956). | Zbl | MR

[24] M. Tsuji, On the uniform distribution of numbers (mod. 1). J. Math. Soc. Japan 4, 313-322 (1952). | Zbl | MR

[25] For references cf. Dr. Cigler'S third paper in this vol. (p. 44). | Numdam

[26] N.M. Koroboff, Einige Probleme der Verteilung von Bruchteilen. Uspechi mat. Nauk 4, 189 -190 (1949).

[27] W. Leveque, On uniform distribution modulo a subdivision. Pacific J. of Math. 8, 757-771 (1953). | Zbl | MR

[28] 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 | MR

[29] 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.

[30] 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

[31] J.F. Koksma, Some theorems on Diophantine inequalities. Scriptum 5 of the Mathematical Centre, Amsterdam (1950). | Zbl | MR

[32] Cf. D. A. Ch. VIII, IX.

[33] J.W.S. Cassels, A new inequality with application to the theory of diophantine approximation. Math. Ann. 126, 108 -118 (1953). | Zbl

[35] Cf. D. A. IX, § 6, p. 116.

Similar problems for generalized dyadic fractions have been treated by C. Sanders, Verdelingsproblemen bij gegeneraliseerde duale breuken. (Diss. Amsterdam V.U.), 1950.

[36] A. Khintchine, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Ergebnisse der Mathematik II, 4, (1933). | JFM

[37] Cf. [36] and e.g. W. Feller, An introduction to probability theory and its applications I, sec. ed. New York-London (1960). | Zbl

[38] 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 | MR

[39] 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

[40] A. Khintchine, Eine arithmetische Eigenschaft der summierbaren Funktionen. Recueil Math., Moscou 41, 11-13 (1934). | Zbl | JFM

[41] C. Ryll-Nardzewski, On the ergodic theorems, I, II. Studia Mathematica XII, 65-79 (1951). | Zbl | MR

[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

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).

[44] Cf. my paper: Sur les suites (λn x) et les fonctions g(t) ∈ L(2). J. de Math. p. appl. 85, 289 - 296 (1956). | Zbl