Problems in Strong Uniform Distribution

[1] BAKER, R. C.: Riemann sums and Lebesgue integrals, Q. J.Math. Oxf. II. Ser. 27 (1976), 191-198.10.1093/qmath/27.2.191Search 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.1909.135.189Search 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/BF02776301Search 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), 248-251.10.2307/1968429Search in Google Scholar

[8] KHINCHIN, A.: Ein Satz ¨uber kettenbr¨uche arithmetichen anwendungen, Math. Z. 18 (1923), 289-306.10.1007/BF01192408Search 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/s3-21.3.540Search in Google Scholar

[11] NAIR, R.: On strong uniform distribution, Acta Arith. 56 (1990), 183-193.10.4064/aa-56-3-183-193Search 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/47.1.101Search in Google Scholar

[13] NAIR, R.: On strong uniform distribution. II., Monatsh. Math. 132 (2001), 341-348.10.1007/PL00010091Search in Google Scholar

[14] NAIR, R.: On a problem of R. C. Baker, Acta Arith. 109 (2003), 343-348.10.4064/aa109-4-4Search in Google Scholar

[15] NAIR, R.: On strong uniform distribution III., Indag. Math. (N.S.) 14 (2003), 233-240.10.1016/S0019-3577(03)90007-3Search 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/CBO9780511721274.016Search 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/BF02566244Search 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-XSearch 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/S0143385709000054Search in Google Scholar

[23] SAWYER, S.: Maximal inequalities of weak type, Ann. of Math. (2) 84 (1966), 157-174.10.2307/1970516Search 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, 1982.10.1007/978-1-4612-5775-2Search in Google Scholar

[26] WEYL, H.: ¨Uber der Gibbssche Erscheinung und verwandte Konvergenzph¨anomenene, Rend. Mat. Palermo 30 (1910), 377-407.10.1007/BF03014883Search in Google Scholar

[27] WEYL, H.: ¨Uber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352.10.1007/BF01475864Search in Google Scholar