In this paper we show discrepancy bounds for index-transformed uniformly distributed sequences. From a general result we deduce very tight lower and upper bounds on the discrepancy of index-transformed van der Corput-, Halton-, and $(t,s)$-sequences indexed by the sum-of-digits function. We also analyze the discrepancy of sequences indexed by other functions, such as, e.g., $\lfloor n^{\alpha}\rfloor$ with $0 < \alpha < 1$.
References
D. Bilyk, M.T. Lacey and A. Vagharshakyan, On the small ball inequality in all dimensions, J. Funct. Anal. 254 (2008), 2470–2502. MR2409170 10.1016/j.jfa.2007.09.010 D. Bilyk, M.T. Lacey and A. Vagharshakyan, On the small ball inequality in all dimensions, J. Funct. Anal. 254 (2008), 2470–2502. MR2409170 10.1016/j.jfa.2007.09.010
J. Dick and F. Pillichshammer, Digital Nets and Sequences–-Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010. MR2683394 J. Dick and F. Pillichshammer, Digital Nets and Sequences–-Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010. MR2683394
M. Drmota and G. Larcher, The sum-of-digits-function and uniform distribution modulo 1, J. Number Theory 89 (2001), 65–96. MR1838704 10.1006/jnth.2000.2628 M. Drmota and G. Larcher, The sum-of-digits-function and uniform distribution modulo 1, J. Number Theory 89 (2001), 65–96. MR1838704 10.1006/jnth.2000.2628
H. Faure and C. Lemieux, Improved Halton sequences and discrepancy bounds, Monte Carlo Models and Applications 16 (2010), 231–250. MR2747814 10.1515/mcma.2010.008 H. Faure and C. Lemieux, Improved Halton sequences and discrepancy bounds, Monte Carlo Models and Applications 16 (2010), 231–250. MR2747814 10.1515/mcma.2010.008
J.H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2 (1960), 84–90. Erratum, ibid. 2: p. 196, 1960. MR121961 10.1007/BF01386213 J.H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2 (1960), 84–90. Erratum, ibid. 2: p. 196, 1960. MR121961 10.1007/BF01386213
P. Hellekalek, A general discrepancy estimate based on $p$-adic arithmetics, Acta Arith. 139 (2009), 117–129. MR2539541 10.4064/aa139-2-3 P. Hellekalek, A general discrepancy estimate based on $p$-adic arithmetics, Acta Arith. 139 (2009), 117–129. MR2539541 10.4064/aa139-2-3
P. Hellekalek, A notion of diaphony based on $p$-adic arithmetic, Acta Arith. 145 (2010), 273–284. MR2733089 10.4064/aa145-3-5 P. Hellekalek, A notion of diaphony based on $p$-adic arithmetic, Acta Arith. 145 (2010), 273–284. MR2733089 10.4064/aa145-3-5
P. Hellekalek, Assessing randomness: tools from $b$-adic analysis, talk at the Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Sydney, Feb. 14 2012. P. Hellekalek, Assessing randomness: tools from $b$-adic analysis, talk at the Tenth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Sydney, Feb. 14 2012.
P. Hellekalek and H. Niederreiter, Constructions of uniformly distributed sequences using the $b$-adic method, Unif. Distrib. Theory 6 (2011), 185–200. MR2817766 P. Hellekalek and H. Niederreiter, Constructions of uniformly distributed sequences using the $b$-adic method, Unif. Distrib. Theory 6 (2011), 185–200. MR2817766
R. Hofer, P. Kritzer, G. Larcher and F. Pillichshammer, Distribution properties of generalized van der Corput-Halton sequences and their subsequences, Int. J. Number Theory 5 (2009), 719–746. MR2532267 10.1142/S1793042109002328 R. Hofer, P. Kritzer, G. Larcher and F. Pillichshammer, Distribution properties of generalized van der Corput-Halton sequences and their subsequences, Int. J. Number Theory 5 (2009), 719–746. MR2532267 10.1142/S1793042109002328
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley, New York, 1974; reprint, Dover Publications, Mineola, NY, 2006. MR419394 L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley, New York, 1974; reprint, Dover Publications, Mineola, NY, 2006. MR419394
A.M. Odlyzko and L.B. Richmond, On the unimodality of high convolutions of discrete distributions, Ann. Probab. 13 (1985), 299–306. MR770644 10.1214/aop/1176993082 euclid.aop/1176993082
A.M. Odlyzko and L.B. Richmond, On the unimodality of high convolutions of discrete distributions, Ann. Probab. 13 (1985), 299–306. MR770644 10.1214/aop/1176993082 euclid.aop/1176993082
K.F. Roth, On irregularities of distribution, Mathematika 1 (1954), 73–79. MR66435 10.1112/S0025579300000541 K.F. Roth, On irregularities of distribution, Mathematika 1 (1954), 73–79. MR66435 10.1112/S0025579300000541
R.F. Tichy and G. Turnwald, Gleichmässige Diskrepanzabschätzung für Ziffernsummen, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 123 (1987), 17–21. MR911762 R.F. Tichy and G. Turnwald, Gleichmässige Diskrepanzabschätzung für Ziffernsummen, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 123 (1987), 17–21. MR911762
R.F. Tichy and G. Turnwald, On the discrepancy of some special sequences, J. Number Theory 26 (1987), 351–366. MR883534 10.1016/0022-314X(87)90096-5 R.F. Tichy and G. Turnwald, On the discrepancy of some special sequences, J. Number Theory 26 (1987), 351–366. MR883534 10.1016/0022-314X(87)90096-5
