Geodesic
On a bounded remainder set for $(t,s)$ sequences~I
Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 224-247.

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Let $\mathbf{x}_0,\mathbf{x}_1,\dots$ be a sequence of points in $[0,1)^s$. A subset $S$ of $[0,1)^s$ is called a bounded remainder set if there exist two real numbers $a$ and $C$ such that, for every integer $N$, $$ | \mathrm{card}\{n \; | \; \mathbf{x}_{n} \in S\} - a N| . $$ Let $ (\mathbf{x}_n)_{n \geq 0} $ be an $s-$dimensional Halton-type sequence obtained from a global function field, $b \geq 2$, $\mathbf{\gamma} =(\gamma_1,...,\gamma_s)$, $\gamma_i \in [0, 1)$, with $b$-adic expansion $\gamma_i= \gamma_{i,1}b^{-1}+ \gamma_{i,2}b^{-2}+...$, $i=1,...,s$. In this paper, we prove that $[0,\gamma_1) \times ...\times [0,\gamma_s)$ is the bounded remainder set with respect to the sequence $(\mathbf{x}_n)_{n \geq 0}$ if and only if \begin{equation} \nonumber \max_{1 \leq i \leq s} \max \{ j \geq 1 \; | \; \gamma_{i,j} \neq 0 \} \infty. \end{equation} We also obtain the similar results for a generalized Niederreiter sequences, Xing-Niederreiter sequences and Niederreiter-Xing sequences.
Keywords: bounded remainder set, $(t,s)$ sequence, Halton type sequences. 
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}
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Mordechay B. Levin. On a bounded remainder set for $(t,s)$ sequences~I. Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 224-247. http://geodesic.mathdoc.fr/item/CHEB_2019_20_1_a13/

[1] Beck J., Chen W. W. L., Irregularities of Distribution, Cambridge Univ. Press, Cambridge, 1987 | MR | Zbl

[2] Bilyk D., “On Roth's orthogonal function method in discrepancy theory”, Unif. Distrib. Theory, 6:1 (2011), 143–184 | MR | Zbl

[3] Dick J., Pillichshammer F., Digital Nets and Sequences, Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010 | MR | Zbl

[4] Grepstad S., Lev N., “Sets of bounded discrepancy for multi-dimensional irrational rotation”, Geom. Funct. Anal., 25:1 (2015), 87–133 | MR | Zbl

[5] Hellekalek P., “Regularities in the distribution of special sequences”, J. Number Theory, 18:1 (1984), 41–55 | MR | Zbl

[6] Larcher, G., “Digital Point Sets: Analysis and Applications”, Springer Lecture Notes in Statistics, 138, 1998, 167–222 | MR | Zbl

[7] Levin M. B., “Adelic constructions of low discrepancy sequences”, Online J. Anal. Comb., 2010, no. 5, 27 pp. | MR

[8] Levin M. B., “On the lower bound of the discrepancy of $(t,s)$ sequences: II”, Online J. Anal. Comb., 2017, no. 5, 74 pp. | MR

[9] Levin M. B., On a bounded remainder set for a digital Kronecker sequence, arXiv: 1901.00042

[10] Levin M. B., On a bounded remainder set for $(t,s)$ sequences II, in preparation

[11] Niederreiter H., Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF Regional Conference Series in Applied Mathematics, 63, SIAM, 1992 | MR | Zbl

[12] Niederreiter H., Yeo A. S., “Halton-type sequences from global function fields”, Sci. China Math., 56 (2013), 1467–1476 | MR | Zbl

[13] Salvador G. D. V., Topics in the Theory of Algebraic Function Fields. Mathematics: Theory $\$ Applications, Birkhauser Boston, Inc., Boston, MA, 2006 | MR

[14] Stichtenoth H., Algebraic Function Fields and Codes, 2nd ed., Springer, Berlin, 2009 | MR | Zbl

[15] Tezuka S., “Polynomial arithmetic analogue of Halton sequences”, ACM Trans Modeling Computer Simulation, 3 (1993), 99–107 | Zbl