On the Small Ball Inequality in all dimensions

Bilyk, Dmitriy; Lacey, Michael T.; Vagharshakyan, Armen

doi:10.1016/j.jfa.2007.09.010

Cited by 95 publications

(108 citation statements)

References 12 publications

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“…There are results for the L 1 ([0, 1) d )-and the star (L ∞ ([0, 1) d )-) discrepancy though there are still gaps between lower and upper bounds, see [H81], [S72], [BLV08]. As general references for studies of the discrepancy function we refer to the monographs [DP10], [NW10], [M99], [KN74] and surveys [B11], [Hi14], [M13c].…”

Section: Introduction and Resultsmentioning

confidence: 99%

L_p- and S_p,q^rB-discrepancy of (order 2) digital nets

Markhasin¹

2015

Acta Arith.

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Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the L p -discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range and enlarge that range for order 2 digitals nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness is considered as well.

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Section: Introduction and Resultsmentioning

confidence: 99%

L_p- and S_p,q^rB-discrepancy of (order 2) digital nets

Markhasin¹

2015

Acta Arith.

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“…, where µ(d) > 0 is some small but strictly positive function of d [9]. For the 2 -discrepancy, the highest lower bound is disc 2 (P, [22,11].…”

Section: Previous Resultsmentioning

confidence: 99%

On Range Searching in the Group Model and Combinatorial Discrepancy

Larsen¹

2014

SIAM J. Comput.

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Abstract. In this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that tutq = Ω(disc 2 ) where tu is the worst case update time, tq the worst case query time and disc is the combinatorial discrepancy of the range searching problem in question. This relation immediately implies a whole range of exceptionally high and near-tight lower bounds for all of the basic range searching problems. We list a few of them in the following:• For d-dimensional halfspace range searching, we get a lower bound of tutq = Ω(n 1−1/d ). This comes within a lg lg n factor of the best known upper bound.• For orthogonal range searching, we get a lower bound of tutq = Ω(lg d−1 n).• For ball range searching, we get a lower bound of tutq = Ω(n 1−1/d ). We note that the previous highest lower bound for any explicit problem, due to Pǎtraşcu [STOC'07], states that tq = Ω((lg n/ lg(lg n + tu)) 2 ), which does however hold for a less restrictive class of data structures.Our result also has implications for the field of combinatorial discrepancy. Using textbook range searching solutions, we improve on the best known discrepancy upper bound for axis-aligned rectangles in all dimensions d ≥ 3.Key words. range searching, lower bounds, group model, discrepancy, computational geometry AMS subject classifications. 68P05,68Q171. Introduction. Range searching is one of the most fundamental and wellstudied topics in the fields of computational geometry and spatial databases. The input to a range searching problem consists of a set of n geometric objects, most typically points in d-dimensional space, and the goal is to preprocess the input into a data structure, such that given a query range, one can efficiently aggregate information about the input objects intersecting the query range. Some of the most typical types of query ranges are axis-aligned rectangles, halfspaces, simplices and balls.The type of information computed over the input objects intersecting a query range include for instance, counting the number of such objects, reporting them and computing the semi-group or group sum of a set of weights assigned to the objects.In the somewhat related field of combinatorial discrepancy, the focus lies on understanding set systems. In particular, if (Y, A) is a set system, where Y = {1, . . . , n} are the elements and A = {A 1 , . . . , A m } is a family of subsets of Y , then the minimum discrepancy problem asks to find a 2-coloring χ : Y → {−1, +1} of the elements in Y , such that each set in A is colored as evenly as possible, i.e. find χ minimizing disc ∞ (χ, Y, A), where

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“…The state of the art and relevant references may be found in [6,12,25] and the recent papers [1,11]. By (3.18) and Proposition 2.19 (and some extra considerations what happens at the boundary ∂Q n ) it follows that…”

Section: Final Commentsmentioning

confidence: 97%