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The Khinchin inequality and Chen’s theorem. (English. Russian original) Zbl 1278.11077

St. Petersbg. Math. J. 23, No. 4, 761-778 (2012); translation from Algebra Anal. 23, No. 4, 179-204 (2011).
Summary: Chen’s theorem on the mean values of \(L_q\)-discrepancies is one of the basic results in the theory of uniformly distributed point sets. This is a difficult result, based on deep and nontrivial combinatorial arguments (see the papers by Chen and Beck on irregularities of distributions). The paper is aimed at showing that the results of such a type are intimately related to lacunarity and statistical independence of certain function series. In particular, the classical Khinchine inequality for the Rademacher functions is employed to prove an important generalization of Chen’s theorem. In a forthcoming paper, the author will continue the study of the phenomena of lacunarity and statistical independence in the context of the theory of uniformly distributed point sets.

MSC:

11K36 Well-distributed sequences and other variations
11K38 Irregularities of distribution, discrepancy
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References:

[1] József Beck and William W. L. Chen, Irregularities of distribution, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, Cambridge, 1987. · Zbl 0617.10039
[2] W. W. L. Chen, On irregularities of distribution. II, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135, 257 – 279. · Zbl 0533.10045 · doi:10.1093/qmath/34.3.257
[3] William W. L. Chen and Maxim M. Skriganov, Orthogonality and digit shifts in the classical mean squares problem in irregularities of point distribution, Diophantine approximation, Dev. Math., vol. 16, SpringerWienNewYork, Vienna, 2008, pp. 141 – 159. · Zbl 1233.11082 · doi:10.1007/978-3-211-74280-8_7
[4] Henri Faure, Discrépance de suites associées à un système de numération (en dimension \?), Acta Arith. 41 (1982), no. 4, 337 – 351 (French). · Zbl 0442.10035
[5] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372 – 414. · Zbl 0036.03604
[6] Ряды и преобразования Уолша, ”Наука”, Мосцощ, 1987 (Руссиан). Теория и применения. [Тхеоры анд апплицатионс]. Б. Голубов, А. Ефимов, анд В. Скворцов, Щалш сериес анд трансформс, Матхематицс анд иц Апплицатионс (Совиет Сериес), вол. 64, Клущер Ацадемиц Публишерс Гроуп, Дордречт, 1991. Тхеоры анд апплицатионс; Транслатед фром тхе 1987 Руссиан оригинал бы Щ. Р. Щаде.
[7] Jiří Matoušek, Geometric discrepancy, Algorithms and Combinatorics, vol. 18, Springer-Verlag, Berlin, 1999. An illustrated guide. · Zbl 0930.11060
[8] Harald Niederreiter, Nets, (\?,\?)-sequences, and algebraic curves over finite fields with many rational points, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 377 – 386. · Zbl 0899.11038
[9] G. Peshkir and A. N. Shiryaev, Khinchin inequalities and a martingale extension of the sphere of their action, Uspekhi Mat. Nauk 50 (1995), no. 5(305), 3 – 62 (Russian); English transl., Russian Math. Surveys 50 (1995), no. 5, 849 – 904. · Zbl 0860.60018 · doi:10.1070/RM1995v050n05ABEH002594
[10] M. M. Skriganov, Coding theory and uniform distributions, Algebra i Analiz 13 (2001), no. 2, 191 – 239 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 2, 301 – 337.
[11] M. M. Skriganov, Harmonic analysis on totally disconnected groups and irregularities of point distributions, J. Reine Angew. Math. 600 (2006), 25 – 49. · Zbl 1115.11047 · doi:10.1515/CRELLE.2006.085
[12] Многомерные квадратурные формулы и функции Хаара., Издат. ”Наука”, Мосцощ, 1969 (Руссиан). · Zbl 0195.16903
[13] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
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