Problems in Strong Uniform Distribution

Chan, Kwo; Nair, Radhakrishnan

doi:10.2478/tmmp-2014-0018

Abstract

In 1923 A. Khinchin asked if given any B ⊆ [0, 1) of positive Lebesgue measure, we have #{n : 1 ≤ n ≤ N : {nx} ∈ B} → |B| for almost all x with respect to Lebesgue measure. Here {y} denotes the fractional part of the real number y and |A| denotes the Lebesgue measure of the set A in [0, 1). In 1970 J. Marstrand showed the answer is no. In this paper the authors survey contributions to this subject since then.

References

[1] BAKER, R. C.: Riemann sums and Lebesgue integrals, Q. J.Math. Oxf. II. Ser. 27 (1976), 191-198.10.1093/qmath/27.2.191
Search in Google Scholar
[2] BEWLEY, T.: Extension of the Birkhoff and von Neumann ergodic theorems to semigroup actions, Ann. Inst. Henri Poincar´e, Nouv. S´er., Sect. B 7 (1971), 283-291.
Search in Google Scholar
[3] BOHL, P.: ¨Uber ein der Theorie der s¨aku¨aren St¨orungen vorkommendes Problem, J. Reine Angew. Math. 135 (1909), 189-289.10.1515/crll.1909.135.189
Search in Google Scholar
[4] BOURGAIN, J.: On maximal ergodic theory for certain subsets of the integers, Isr. J. Math. 61 (1988), 39-72.10.1007/BF02776301
Search in Google Scholar
[5] BOURGAIN, J.: Almost sure convergence in ergodic theory, in: Proc. Int. Conf., Almost Everywhere Convergence, Columbus, OH, 1988, Acadamic, Boston, 1989, pp. 145-151.
Search in Google Scholar
[6] CHAN, K.-NAIR, R.: A remark on the distribution of Chebychev polynomials on [-1,1], Unif. Distrib. Theory (to appear).
Search in Google Scholar
[7] JESSEN, B.: On the approximation of Lebesgue integrals by Riemann sums, Ann. Math. (2) 35 (1934), 248-251.10.2307/1968429
Search in Google Scholar
[8] KHINCHIN, A.: Ein Satz ¨uber kettenbr¨uche arithmetichen anwendungen, Math. Z. 18 (1923), 289-306.10.1007/BF01192408
Search in Google Scholar
[9] KOKSMA, J. F.-SALEM, R.: Uniform distribution and Lebesgue integration, Acta Sci. Math. (Szeged) 12 (1950); Actual. Math. Centrum, Amsterdam, Rapport ZW 1949, 004, 9 p.
Search in Google Scholar
[10] MARSTRAND, J. M.: On Khinchine’s conjecture about strong uniform distribution, Proc. Lond. Math. Soc. 21 (1970), 540-556.10.1112/plms/s3-21.3.540
Search in Google Scholar
[11] NAIR, R.: On strong uniform distribution, Acta Arith. 56 (1990), 183-193.10.4064/aa-56-3-183-193
Search in Google Scholar
[12] NAIR, R.: On some arithmetic properties of Lp summable functions, Q. J. Math. Oxf. II. Ser. 47 (1996), 101-105.10.1093/qmath/47.1.101
Search in Google Scholar
[13] NAIR, R.: On strong uniform distribution. II., Monatsh. Math. 132 (2001), 341-348.10.1007/PL00010091
Search in Google Scholar
[14] NAIR, R.: On a problem of R. C. Baker, Acta Arith. 109 (2003), 343-348.10.4064/aa109-4-4
Search in Google Scholar
[15] NAIR, R.: On strong uniform distribution III., Indag. Math. (N.S.) 14 (2003), 233-240.10.1016/S0019-3577(03)90007-3
Search in Google Scholar
[16] NAIR, R.: On strong uniform distribution IV., J. Inequal. Appl. 2005, no. 3, 319-327.
Search in Google Scholar
[17] NAIR, R.: On polynomial ergodic averages and square functions, in: Number Theory and Polynomials. Proc. of the Workshop (J. McKee et al., eds.), Bristol, UK, 2006, London Math. Soc. Lecture Note Ser., Vol. 352, Cambridge University Press, Cambridge, 2008, pp. 241-254.10.1017/CBO9780511721274.016
Search in Google Scholar
[18] RAIKOV, D. A.: On some arithmetic properties of summable functions, Mat. Sb. 1 (43), (1936), 377-384. (In Russian)
Search in Google Scholar
[19] REINHOLD-LARSSON, K.: Discrepancy of behaviour of perturbed sequences in Lp spaces, Proc. Amer. Math. Soc. 120 (1994), 865-874.
Search in Google Scholar
[20] REISZ, F.: Sur la th´eorie ergodique, Comment. Math. Helv. 17 (1944/45), 221-239.10.1007/BF02566244
Search in Google Scholar
[21] ROSENBLATT, J.: Universally bad sequences in ergodic theory, in: Almost everywhere convergence II, Proc. 2nd Int. Conf., Evanston/IL (USA) 1989, Academic Press, Boston, MA, 1991, pp. 227-245.10.1016/B978-0-12-085520-9.50023-X
Search in Google Scholar
[22] QUAS, A.-WIERDL, M.: Rates of divergence of non-conventional ergodic averages, Ergodic Theory Dynam. Systems 30 (2010), 233-262.10.1017/S0143385709000054
Search in Google Scholar
[23] SAWYER, S.: Maximal inequalities of weak type, Ann. of Math. (2) 84 (1966), 157-174.10.2307/1970516
Search in Google Scholar
[24] SIERPIŃSKI, W.: On the asymptotic value of a certain sum, Rozprawy Wydz. Mat. Pryzr. Akad. Um. 50 (1910), 1-10. (In Polish)
Search in Google Scholar
[25] WALTERS, P.: An Introduction to Ergodic Theory, in: Grad. Texts in Math. Vol. 79, Springer-Verlag, Berlin, 1982.10.1007/978-1-4612-5775-2
Search in Google Scholar
[26] WEYL, H.: ¨Uber der Gibbssche Erscheinung und verwandte Konvergenzph¨anomenene, Rend. Mat. Palermo 30 (1910), 377-407.10.1007/BF03014883
Search in Google Scholar
[27] WEYL, H.: ¨Uber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352.10.1007/BF01475864
Search in Google Scholar

Problems in Strong Uniform Distribution

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