On the Intriguing Search for Good Permutations

[1] BEHNKE, H.: Zur Theorie der diophantischen Approximationen, Abh. Math.Sem.Hamburg 3 (1924), 261–318; 4 (1926), 33–46.Search in Google Scholar

[2] BÉJIAN, R.: Minoration de la discrépance d’une suite quelconque sur T,Acta Arith. 41 (1982), 185–202.10.4064/aa-41-2-185-202Search in Google Scholar

[3] BÉJIAN, R.—FAURE, H.: Discrépance de la suite de van der Corput, C.R. Acad. Sci. Paris Ser. A 285 (1977), 313–316.Search in Google Scholar

[4] BOURGAIN, J.—KONTOROVICH, A.: On Zaremba’s conjecture, Ann. of Math. (2) 180 (2014), 1–60.10.4007/annals.2014.180.1.3Search in Google Scholar

[5] CARLITZ, L.: Permutations in a finite field. Proc. Amer. Math. Soc. 4 (1953), 538.10.1090/S0002-9939-1953-0055965-8Search in Google Scholar

[6] CHAIX, H.—FAURE, H.: Discrépance et diaphonie en dimension un,Acta Arith. 63 (1993), 103–141.10.4064/aa-63-2-103-141Search in Google Scholar

[7] DE CLERCK, L.: A method for exact calculation of the star discrepancy of plane sets applied to sequences of Hammersley, Monatsh. Math. 101 (1986), 261–278.10.1007/BF01559390Search in Google Scholar

[8] DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511761188Search in Google Scholar

[9] DRMOTA, M.—TICHY, R.F.: Sequences, Discrepancies and Applications. Lecture Notes in Mathematics Vol. 1651. Springer Verlag, Berlin, 1997.Search in Google Scholar

[10] DUPAIN, Y.—SÓS, V.: On the discrepancy of (nα) sequences. Topics in classical number theory, Vol. I, II (Budapest, 1981), Colloq. Math. Soc. Janos Bolyai 34 (1984), 355–387.Search in Google Scholar

[11] FAURE, H.: Discrépance de suites associées à un système de numération (en dimension un), Bull. Soc. Math. France 109 (1981), 143–182.10.24033/bsmf.1935Search in Google Scholar

[12] –––––– On the star-discrepancy of generalized Hammersley sequences in two dimensions, Monatsh. Math. 101 (1986), 291–300.10.1007/BF01559392Search in Google Scholar

[13] FAURE, H.: Good permutations for extreme discrepancy, J. Number Theory 42 (1992), 47–56.10.1016/0022-314X(92)90107-ZSearch in Google Scholar

[14] –––––– Discrepancy and diaphony of digital (0, 1)-sequences in prime base,Acta Arith. 117 (2005), 125–148.10.4064/aa117-2-2Search in Google Scholar

[15] –––––– Irregularities of distribution of digital (0, 1)-sequences in prime base, Integers 5 (2005), no. 3, #A07.Search in Google Scholar

[16] –––––– Selection criteria for (random) generation of digital (0,s)-sequences. In: Proceedings of the 6th international conference Monte Carlo and quasi-Monte Carlo methods in scientific computing Juan-les-Pins, France, July 7–10, 2004, Monte Carlo and Quasi-Monte Carlo Methods 2004 (H. Niederreiter and D. Talay, eds.). Springer Berlin (2006), pp. 113–126.Search in Google Scholar

[17] –––––– Van der Corput sequences towards (0, 1)-sequences in base b,J. Théor. Nombres Bordeaux 19 (2007), 125–140.10.5802/jtnb.577Search in Google Scholar

[18] –––––– Star extreme discrepancy of generalized two-dimensional Hammersley point sets, Unif. Distrib. Theory 3 (2008), 45–65.Search in Google Scholar

[19] FAURE, H.—KRITZER, P.—PILLICHSHAMMER, F.: From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules, Indag. Math., New Ser. 26 (2015), 760–822.Search in Google Scholar

[20] FAURE, H.—LEMIEUX, C.: Generalized Halton sequences in 2008: A comparative study, ACM Trans. Model. Comp. Sim. 19 (2009), 15:1-31.10.1145/1596519.1596520Search in Google Scholar

[21] FAURE, H.—PILLICHSHAMMER, F.: L_pdiscrepancy of generalized two-dimensional Hammersley point sets, Monatsh. Math. 158 (2009), 31–61.10.1007/s00605-008-0039-1Search in Google Scholar

[22] FAURE, H.—PILLICHSHAMMER, F.: L²discrepancy of two dimensional digitally shifted Hammersley point sets in base b. In: Proceedings of the 8th international conference Monte Carlo and quasi-Monte Carlo methods in scientific computing, Montréal, Canada, July 6–11, 2008, Monte Carlo and quasi-Monte Carlo methods 2008 (P. L’Ecuyer and A. B. Owen, eds.) Springer Berlin (2009), pp. 355–368.Search in Google Scholar

[23] FAURE, H.—PILLICHSHAMMER, F.—PIRSIC, G.—SCHMID, W. CH.: L²discrepancy of generalized two dimensional Hammersley point sets scrambled with arbitrary permutations,Acta Arith. 141 (2010), 395–418.10.4064/aa141-4-6Search in Google Scholar

[24] FAURE, H.—PILLICHSHAMMER, F.—PIRSIC, G.: L²discrepancy of linearly digit scrambled Zaremba point sets, Unif. Distrib. Theory 6 (2011), 59–81.Search in Google Scholar

[25] FAURE, H.—PILLICHSHAMMER, F.: L²discrepancy of generalized Zaremba point sets, J. Théor. Nombres Bordeaux 23 (2011), 121–136.10.5802/jtnb.753Search in Google Scholar

[26] –––––– A generalization of NUT digital (0, 1)-sequences and best possible lower bounds for star discrepancy,Acta Arith. 158 (2013), 321–340.10.4064/aa158-4-2Search in Google Scholar

[27] HABER, S.: On a sequence of points of interest for numerical quadrature,J. Res. Nat. Bur. Stand. Sect. B 70B (1966), no. 2. 127–136.Search in Google Scholar

[28] HALTON, J. H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2 (1960), 84–90.Search in Google Scholar

[29] HALTON, J. H.—ZAREMBA, S. K.: The extreme and the L²discrepancies of some plane sets, Monatsh. Math. 73 (1969), 316–328.10.1007/BF01298982Search in Google Scholar

[30] HAMMERSLEY, J. M.: Monte Carlo methods for solving multivariable problems, Ann. New York Acad. Sci. 86 (1960), 844–874.10.1111/j.1749-6632.1960.tb42846.xSearch in Google Scholar

[31] HUANG, S.: An improvement to Zaremba’s conjecture, Geom. Funct. Anal. 25 (2015), no. 3, 860–914.Search in Google Scholar

[32] KHINCHIN, A. YA. : Continued Fractions. the University of Chicago Press, Chicago, Ill.-London 1964.10.1063/1.3051235Search in Google Scholar

[33] KRITZER. P.: A new upper bound on the star discrepancy of (0, 1)-sequences, Integers 5(3) (2005), A11.Search in Google Scholar

[34] –––––– On some remarkable properties of the two-dimensional Hammersley point set in base 2,J. Théor. Nombres Bordeaux 18 (2006), 203–221.10.5802/jtnb.540Search in Google Scholar

[35] KRITZER. P.—LARCHER, G.—PILLICHSHAMMER, F.: A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets, Ann. Mat. Pura Appl. 186 (2007), 229–250.10.1007/s10231-006-0002-5Search in Google Scholar

[36] KUIPERS, L.—NIEDERREITER, H.: Uniform Distribution of Sequences. Wiley, New York, 1974.Search in Google Scholar

[37] LARCHER, G.: On the star-discrepancy of sequences in the unit-interval,J. Complexity 31 (2015), 474–485.10.1016/j.jco.2014.07.005Search in Google Scholar

[38] –––––– On the discrepancy of sequences in the unit-interval, Indag. Math. (N.S.) 27 (2016), 546–558.10.1016/j.indag.2015.11.003Search in Google Scholar

[39] LARCHER, G.—PUCHHAMMER, F.: An improved bound for the star discrepancy of sequences in the unit interval, Unif. Distrib. Theory 11 (2016), 1–14.10.1515/udt-2016-0001Search in Google Scholar

[40] –––––– Sums of distances to the nearest integer and the discrepancy of digital nets,Acta Arith. 106.4 (2003), 379–408.10.4064/aa106-4-4Search in Google Scholar

[41] MATOUŠEK, J.: On the L²-discrepancy for anchored boxes,J. Complexity 14 (1998), 527–556.10.1006/jcom.1998.0489Search in Google Scholar

[42] NIEDERREITER, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, 1992.10.1137/1.9781611970081Search in Google Scholar

[43] OSTROMOUKHOV, V.: Recent progress in improvement of extreme discrepancy and star discrepancy of one-dimensional sequences. In: Monte Carlo and Quasi-Monte Carlo Methods 2008 (L’Ecuyer, P. and A. B. Owen, eds.), Springer-Verlag, Berlin, 2009, 561–572.10.1007/978-3-642-04107-5_36Search in Google Scholar

[44] OSTROWSKI, A.: Bemerkungen zur Theorie der Diophantischen Approximationen I, II, III,Abh. Math.Sem.Hamburg 1 (1922), no. 1, 77–98, 249–250; 4 (1925), no. 1, 224.Search in Google Scholar

[45] PROINOV, P. D.: Estimation of L²-discrepancy of a class of infinite sequences, C.R. Acad. Bulgare Sci. 36 (1983), 37–40.Search in Google Scholar

[46] –––––– On irregularities of distribution. C.R. Acad Bulgare Sci. 39 (1986), 31–34.Search in Google Scholar

[47] PROINOV, P. D.—ATANASSOV, E, Ẏ.: On the distribution of the van der Corput generalized sequence, C.R. Acad. Sci. Paris Sér. I Math. 307 (1988), 895–900.Search in Google Scholar

[48] PROINOV, P. D.—GROZDANOV, V. S.: On the diaphony of the van der Corput-Halton sequence, J. Number Theory 30 (1988), 94–104.10.1016/0022-314X(88)90028-5Search in Google Scholar

[49] PAUSINGER, F.: Weak multipliers for generalized van der Corput sequences,J. Théor. Nombres Bordeaux 24 (2012), no. 3, 729–749.Search in Google Scholar

[50] PAUSINGER, F.—SCHMID, W. CH. : A good permutation for one-dimensional diaphony, Monte Carlo Methods Appl. 16 (2010), 307–322.10.1515/mcma.2010.015Search in Google Scholar

[51] F. PAUSINGER — A. TOPUZOĞLU: On the discrepancy of two families of permuted van der Corput sequences, Unif. Distrib. Theory 13 (2018), 47–64.10.1515/udt-2018-0003Search in Google Scholar

[52] PILLICHSHAMMER, F.: On the discrepancy of (0, 1)-sequences, J. Number Theory 104 (2004), 301–314.10.1016/j.jnt.2003.08.002Search in Google Scholar

[53] RAMSHAW, L.: On the discrepancy of the sequence formed by the multiples of an irrational number, J. Number Theory 30 (1981), 138–175.10.1016/0022-314X(81)90002-0Search in Google Scholar

[54] ROTH, K. F.: On irregularities of distribution,Mathematika 1 (1954), 73–79.10.1112/S0025579300000541Search in Google Scholar

[55] SCHMIDT, W. M.: Irregularities of distribution VII,Acta Arith. 21 (1972), 45–50.10.4064/aa-21-1-45-50Search in Google Scholar

[56] TOPUZOĞLU, A.: The Carlitz rank of permutations of finite fields: a survey,J.Symb. Comput. 64 (2014), 53–66.10.1016/j.jsc.2013.07.004Search in Google Scholar

[57] VAN DER CORPUT, J. G.: Verteilungsfunktionen. I, Proc. Kon. Ned. Akad. v. Wetensch. 38 (1935), 813–821.Search in Google Scholar

[58] –––––– Verteilungsfunktionen. II, Proc. Kon. Ned. Akad. v. Wetensch. 38 (1935), 1058–1066.Search in Google Scholar

[59] WEYL, H.:Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.10.1007/BF01475864Search in Google Scholar

[60] XIAO, Y. J.: Suitesé quiréparties associées aux automorphismes du tore. C.R. Acad. Sci. Paris Sér. I Math. 311 (1990), 579–582.Search in Google Scholar

[61] ZAREMBA, S. K.: La méthode des bons treillis pour le calcul des intégrals multiples. In:Proc. Sympos.Univ. Montréal, September 9–14, 1971, Applications of Number Theory to Numerical Analysis (S.K. Zaremba, ed.), Academic Press, New York (1972), pp. 39–119.Search in Google Scholar

[62] ZINTERHOF, P.:Über einige Absch¨atzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden,Österr. Akad. Wiss. SB II 185 (1976), 121–132.Search in Google Scholar