On the lower bound of the discrepancy of (t, s)-sequences: II

Mordechay B. Levin

On the lower bound of the discrepancy of (t, s)-sequences: II

Mordechay B. Levin

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let (x(n))_n≥1 be an s−dimensional Niederreiter-Xing’s sequence in base b. Let D((x(n))_n=1^N) be the discrepancy of the sequence (x(n))_n=1^N. It is known that ND((x(n))_n=1^N) = O(ln^s N) as N → ∞. In this paper, we prove that this estimate is exact. Namely, there exists a constant K > 0, such that inf_w∈[0,1)s sup_1≤N≤bm ND((x(n) ⊕ w)_n=1^N) ≥ Km^s for m = 1, 2, ... . We also get similar results for other explicit constructions of (t, s)-sequences.

Original language	English
Journal	Online Journal of Analytic Combinatorics
Issue number	12
State	Published - 2017

Bibliographical note

Publisher Copyright:
© 2017, Department of Computer Science. All rights reserved.

Keywords

(t, s)-sequences
(t,m, s)-nets
Low discrepancy sequences

Cite this

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abstract = "Let (x(n))n≥1 be an s−dimensional Niederreiter-Xing{\textquoteright}s sequence in base b. Let D((x(n))n=1N) be the discrepancy of the sequence (x(n))n=1N. It is known that ND((x(n))n=1N) = O(lns N) as N → ∞. In this paper, we prove that this estimate is exact. Namely, there exists a constant K > 0, such that infw∈[0,1)s sup1≤N≤bm ND((x(n) ⊕ w)n=1N) ≥ Kms for m = 1, 2, ... . We also get similar results for other explicit constructions of (t, s)-sequences.",

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N2 - Let (x(n))n≥1 be an s−dimensional Niederreiter-Xing’s sequence in base b. Let D((x(n))n=1N) be the discrepancy of the sequence (x(n))n=1N. It is known that ND((x(n))n=1N) = O(lns N) as N → ∞. In this paper, we prove that this estimate is exact. Namely, there exists a constant K > 0, such that infw∈[0,1)s sup1≤N≤bm ND((x(n) ⊕ w)n=1N) ≥ Kms for m = 1, 2, ... . We also get similar results for other explicit constructions of (t, s)-sequences.

AB - Let (x(n))n≥1 be an s−dimensional Niederreiter-Xing’s sequence in base b. Let D((x(n))n=1N) be the discrepancy of the sequence (x(n))n=1N. It is known that ND((x(n))n=1N) = O(lns N) as N → ∞. In this paper, we prove that this estimate is exact. Namely, there exists a constant K > 0, such that infw∈[0,1)s sup1≤N≤bm ND((x(n) ⊕ w)n=1N) ≥ Kms for m = 1, 2, ... . We also get similar results for other explicit constructions of (t, s)-sequences.

KW - (t, s)-sequences

KW - (t,m, s)-nets

KW - Low discrepancy sequences

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JO - Online Journal of Analytic Combinatorics

JF - Online Journal of Analytic Combinatorics

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ER -

On the lower bound of the discrepancy of (t, s)-sequences: II

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