TY - GEN
T1 - Tusnády's problem, the transference principle, and non-uniform QMC sampling
AU - Aistleitner, Christoph
AU - Bilyk, Dmitriy
AU - Nikolov, Aleksandar
PY - 2018
Y1 - 2018
N2 - It is well-known 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
-
1N-1. 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 so-called transference principle together with recent results on the discrepancy of red-blue colorings to show that for any μ there even exist points having discrepancy of order at most (logN)d-12N-1, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.
AB - It is well-known 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
-
1N-1. 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 so-called transference principle together with recent results on the discrepancy of red-blue colorings to show that for any μ there even exist points having discrepancy of order at most (logN)d-12N-1, which is almost as good as the discrepancy bound in the case of the Lebesgue measure.
KW - combinatorial discrepancy
KW - Gates of Hell
KW - Low-discrepancy sequences
KW - Non-uniform sampling
KW - Tusnády’s problem
UR - http://www.scopus.com/inward/record.url?scp=85049890770&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-91436-7_8
DO - 10.1007/978-3-319-91436-7_8
M3 - Conference paper
AN - SCOPUS:85049890770
SN - 9783319914350
T3 - Springer Proceedings in Mathematics & Statistics
SP - 169
EP - 180
BT - Monte Carlo and Quasi-Monte Carlo Methods - MCQMC 2016
PB - Springer New York LLC
T2 - 12th International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing
Y2 - 14 August 2016 through 19 August 2016
ER -