A note on isotropic discrepancy and spectral test of lattice point sets

Pillichshammer, Friedrich; Sonnleitner, Mathias

doi:10.1016/j.jco.2019.101441

Cited by 2 publications

(5 citation statements)

References 19 publications

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 96%

“…In our recent paper [13] we exhibited a close connection between the isotropic discrepancy and the spectral test of lattice point sets in the d-dimensional unit cube [0, 1) d . For the definition of these well-established notions we refer to Hellekalek [8] as well as [13]. The central result is [13,Theorem 2] which states that the isotropic discrepancy J N of an N-element lattice point set P(L) in [0, 1) d and the spectral test σ(L) are -up to multiplicative factors only depending on the dimension d -equivalent, i.e., we have J N (P(L)) ≍ d σ(L).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 96%

“…For the definition of these well-established notions we refer to Hellekalek [8] as well as [13]. The central result is [13,Theorem 2] which states that the isotropic discrepancy J N of an N-element lattice point set P(L) in [0, 1) d and the spectral test σ(L) are -up to multiplicative factors only depending on the dimension d -equivalent, i.e., we have J N (P(L)) ≍ d σ(L). Possible choices for the involved multiplicative factors are stated explicitly.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Using a slightly different setup than [13], Aistleitner et al considered translates of fundamental cells which they intersected with the unit square. However, also their approach requires some modification since in [1] the last equality on page 1006 and the first on page 1007 are not correct.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…, where N k = |P(L N k )| and there exist sequences achieving this rate, see, e.g., [13,Propositions 3 and 4]. Thus, we can interpret Theorem 2 in the following way: Corollary 3.…”

Section: Given a Sequence Of Integration Lattices (Lmentioning

confidence: 98%

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On the relation of the spectral test to isotropic discrepancy and $L_q$-approximation in Sobolev spaces

Sonnleitner,

Pillichshammer

2020

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This paper is a follow-up to the recent paper "A note on isotropic discrepancy and spectral test of lattice point sets" [J. Complexity, 58:101441, 2020]. We show that the isotropic discrepancy of a lattice point set is at most d 2 2(d+1) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube.Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test -and with it the isotropic discrepancy -is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 96%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 96%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…, where N k = |P(L N k )| and there exist sequences achieving this rate, see, e.g., [13,Propositions 3 and 4]. Thus, we can interpret Theorem 2 in the following way: Corollary 3.…”

Section: Given a Sequence Of Integration Lattices (Lmentioning

confidence: 98%

See 3 more Smart Citations