The theory of asymptotic distribution modulo one

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1. [1] For references till 1986 cf. my Diophantische Approximationen (Berlin1936, Ergebnisse der MathematikIV, 4), in the following denoted by D. A. JFM62.0173.01
2. [2] J. Cigler und G. Helmberg, Neuere Entwicklungen der Theorie der Gleichverteilung. Jahresbericht der D.M.V.64, 1-50 (1961). Zbl0109.03404MR125102
3. A large part of: J.W.S. Cassels , An introduction to diophantine approximation (Cambridge Univ. Tract45, 1957) also is dedicated to our subject. Zbl0077.04801MR87708
4. [3] E.g. P. Erdös in his contribution to this symposium: Problems and results on diophantine approximations, (this volume p. 52). 
5. [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). 
6. Eenige integralen in de theorie der gelijkmatige verdeeling modulo 1. Mathematica (Zutphen)11B, 49-52 (1942/48). 
7. [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)). Zbl0015.00602
8. B. Eckmann, Über monothetische Gruppen. Comment. math. Helvet.16, 249- 268 (1948/44). Zbl0061.04402MR11302
9. [8] E.g. by L. Kuipers and B. Meulenbeld. For references cf. CIGLER-HELMBERG quoted in [2]. 
10. [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 Mathematicae12, 192-207 (1950). Zbl0041.02406MR36291
11. [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 Mathematicae17, 65-84 (1955). Zbl0068.05403
12. [11] In his paper: Über die Gleichverteilung von Zahlen modulo Eins, Math. Ann.77, 313 - 352 (1916) p. 845. Zbl46.0278.06MR1511862JFM46.0278.06
13. [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. JFM59.0958.02
14. [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). Zbl0001.20102JFM57.0230.05
15. [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). Zbl0208.31002
16. [15] I. Schoenberg, Ueber die asymptotische Verteilung reeller Zahlen mod. 1. Math. Z.28, 171-199 (1928). Zbl54.0212.02MR1544950JFM54.0212.02
17. [18] R.J. Duffin and A.C. Schaeffer, Khintchine's problems in metric Diophantine approximation. Duke Math. J.8, 248-255 (1941). Zbl0025.11002JFM67.0145.03
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 Mathematicae7, 54 - 70 (1945), where also further references are given. Zbl0060.12206MR15096
20. J.W.S. Cassels, Some metrical theorems in diophantine approximation. IProc. Cambr. Phil. Soc.46, 209 - 218 (1949).IIJ. London Math. Soc.25, 180 -184 (1950). Zbl0037.17201MR36787
21. [19] D. De Vries, Metrische onderzoekingen van Diophantische benaderingsproblemen in het niet-lacunaire geval. (Diss. Amsterdam, V.U.), 1955. 
22. [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). Zbl0014.20803JFM62.0207.03
23. [21] For references cf. K. Roth, On irregularities of distribution. Mathematika (London) 1, 73-79 (1954). Zbl0057.28604
24. [22] H. Davenport, Note on irregularities of distribution. Mathematika (London), 3, 131-135 (1956). Zbl0073.03402MR82531
25. [24] M. Tsuji, On the uniform distribution of numbers (mod. 1). J. Math. Soc. Japan4, 313-322 (1952). Zbl0048.03302MR59322
26. [25] For references cf. Dr. Cigler's third paper in this vol. (p. 44). 
27. [26] N.M. Koroboff, Einige Probleme der Verteilung von Bruchteilen. Uspechi mat. Nauk4, 189 -190 (1949). 
28. [27] W. Leveque, On uniform distribution modulo a subdivision. Pacific J. of Math.8, 757-771 (1953). Zbl0051.28503MR59323
29. [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. Zbl0042.28803MR42484
30. [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. AmsterdamV.U.), 1945. 
31. [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 Mathematicae10, 370-378; 406 - 413 (1948). Zbl0031.25402
32. [31] J.F. Koksma, Some theorems on Diophantine inequalities. Scriptum 5 of the Mathematical Centre, Amsterdam (1950). Zbl0038.02803MR38379
33. [32] Cf. D. A. Ch. VIII, IX. 
34. [33] J.W.S. Cassels, A new inequality with application to the theory of diophantine approximation. Math. Ann.126, 108 —118 (1953). Zbl0051.28604
35. [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. AmsterdamV.U.), 1950. 
37. [36] A. Khintchine, Asymptotische Gesetze der Wahrscheinlichkeitsrechnung, Ergebnisse der MathematikII, 4, (1933). Zbl59.1153.01JFM59.1153.01
38. [37] Cf. [36] and e.g. W. Feller, An introduction to probability theory and its applications I, sec. ed.New York-London (1960). Zbl0138.10207
39. [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 Mathematicae17, 64-84 (1955). Zbl0068.05403MR69309
40. [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 Mathematicae12, 354-367 (1950). Zbl0038.19102
41. [40] A. Khintchine, Eine arithmetische Eigenschaft der summierbaren Funktionen. Recueil Math., Moscou41, 11-13 (1934). Zbl0009.30602JFM60.0979.03
42. [41] C. Ryll-Nardzewski, On the ergodic theorems, I, II. Studia MathematicaXII, 65-79 (1951). Zbl0044.12302MR46582
43. [43] A proof of the first counter example in J.F. Koksma- R. Salem, Uniform distribution and Lebesgue integration. Acta Scient. Math. Szeged12, 87-96 (1950). Zbl0036.03101
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). 
45. [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). Zbl0070.28402

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