TY  GEN
T1  Tusnády's problem, the transference principle, and nonuniform QMC sampling
AU  Aistleitner, Christoph
AU  Bilyk, Dmitriy
AU  Nikolov, Aleksandar
PY  2018
Y1  2018
N2  It is wellknown that for every N≥ 1 and d≥ 1 there exist point sets x1, ⋯, xN∈ [0, 1]d whose discrepancy with respect to the Lebesgue measure is of order at most (log N)d

1N1. In a more general setting, the first author proved together with Josef Dick that for any normalized measure μ on [0, 1 ]d there exist points x1, ⋯, xN whose discrepancy with respect to μ is of order at most (log N)(
3
d
+
1
)
/
2N 1. The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the socalled transference principle together with recent results on the discrepancy of redblue colorings to show that for any μ there even exist points having discrepancy of order at most (logN)d12N1, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.
AB  It is wellknown that for every N≥ 1 and d≥ 1 there exist point sets x1, ⋯, xN∈ [0, 1]d whose discrepancy with respect to the Lebesgue measure is of order at most (log N)d

1N1. In a more general setting, the first author proved together with Josef Dick that for any normalized measure μ on [0, 1 ]d there exist points x1, ⋯, xN whose discrepancy with respect to μ is of order at most (log N)(
3
d
+
1
)
/
2N 1. The proof used methods from combinatorial mathematics, and in particular a result of Banaszczyk on balancings of vectors. In the present note we use a version of the socalled transference principle together with recent results on the discrepancy of redblue colorings to show that for any μ there even exist points having discrepancy of order at most (logN)d12N1, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.
KW  combinatorial discrepancy
KW  Gates of Hell
KW  Lowdiscrepancy sequences
KW  Nonuniform sampling
KW  Tusnády’s problem
UR  http://www.scopus.com/inward/record.url?scp=85049890770&partnerID=8YFLogxK
U2  10.1007/9783319914367_8
DO  10.1007/9783319914367_8
M3  Conference paper
AN  SCOPUS:85049890770
SN  9783319914350
T3  Springer Proceedings in Mathematics & Statistics
SP  169
EP  180
BT  Monte Carlo and QuasiMonte Carlo Methods  MCQMC 2016
PB  Springer New York LLC
T2  12th International Conference on Monte Carlo and QuasiMonte Carlo Methods in Scientific Computing
Y2  14 August 2016 through 19 August 2016
ER 