Dynamically defined sequences with small discrepancy

Steinerberger, Stefan

doi:10.1007/s00605-019-01360-z

Cited by 14 publications

(13 citation statements)

References 15 publications

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“…The second statement in the main theorem, i.e. discrepancy being preserved over all possible choices, shows that potential theoretic approaches along the lines of what was proposed by Steinerberger [25,26]…”

Section: Remarkmentioning

confidence: 74%

“…Motivated by this, Steinerberger [25,26] recently proposed to study whether regular sequences could be constructed via dynamical systems of the type outlined in (1). More precisely, suppose we are given {x 0 , … , x N−1 } ⊂ [0, 1) , then he proposed to construct x N in a greedy manner as and if the minimum is not unique, any choice is admissible.…”

Section: A Possible Connectionmentioning

confidence: 99%

“…The functional studied by Steinerberger [25,26] is an example of an admissible function; for x ∈ (0, 1) see Fig. 2 for an illustration.…”

Section: Definition 51 Letmentioning

confidence: 99%

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Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Pausinger

2020

Annali di Matematica

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This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing ’well-distributed’ sequences of points on [0, 1). Let $$f:[0,1] \rightarrow {\mathbb {R}}$$ f : [ 0 , 1 ] → R be (1) symmetric $$f(x) = f(1-x)$$ f ( x ) = f ( 1 - x ) , (2) twice differentiable on (0, 1), and (3) such that $$f''(x)>0$$ f ′ ′ ( x ) > 0 for all $$x \in (0,1)$$ x ∈ ( 0 , 1 ) . We study the greedy dynamical system, where, given an initial set $$\{x_0, \ldots , x_{N-1}\} \subset [0,1)$$ { x 0 , … , x N - 1 } ⊂ [ 0 , 1 ) , the point $$x_N$$ x N is obtained as $$\begin{aligned} x_{N} = \arg \min _{x \in [0,1)} \sum _{k=0}^{N-1}{f(|x-x_k|)}. \end{aligned}$$ x N = arg min x ∈ [ 0 , 1 ) ∑ k = 0 N - 1 f ( | x - x k | ) . We prove that if we start this construction with the single element $$x_0=0$$ x 0 = 0 , then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): greedy energy minimization recovers the way we count in binary. This gives a new construction of the classical van der Corput sequence. The special case $$f(x) = 1-\log (2 \sin (\pi x))$$ f ( x ) = 1 - log ( 2 sin ( π x ) ) answers a question of Steinerberger. Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk.

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Section: Remarkmentioning

confidence: 74%

Section: A Possible Connectionmentioning

confidence: 99%

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Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Pausinger

2020

Annali di Matematica

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“…Götz [25], building on earlier machinery of Andrievski and Blatt [3, 4], Blatt [14], Blatt and Mhaskar [15], and Totik [45], proved that

false∥ D_{N} {false∥}_{L^{\infty}} ≲ N^{- 1 / 2} \log N

. The author (unknowingly) recovered this bound in a different setting [44] (the guiding motivation was to interpret the Erdős–Turan inequality as an energy functional; this perspective was also useful in [43]). A philosophically related object was studied in [2, 27, 28].…”

Section: Statement Of Resultsmentioning

confidence: 99%

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

Steinerberger

2021

Mathematika

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Let z 1 , . . . , z m be m distinct complex numbers, normalized to |z k | = 1, and consider the polynomialWe define a sequence of polynomials in a greedy fashion,whereand prove that, independently of the initial polynomial p m , the roots of p N equidistribute in angle at rate at most (log N ) 2 /N. This persists when sometimes adding "adversarial" points by hand. We also obtain sharp rates for an L 2 -version of a problem first raised by Erdős and solved by Beck in L ∞ . §1. Introduction.1.1. Introduction. Let (x n ) ∞ n=1 be a sequence on [0,1]. We define the discrepancy function D N : [0, 1] → [0, 1] associated of the first N elements via

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“…In particular, we do not claim that our energy functional is necessarily the most effective one. Our functional certainly seems natural in light of our derivation; moreover, the author recently used it [36] to define sequences (x n ) ∞ n=1 whose discrepancy seems to be extremely good when compared to classical sequences (however, the only known bound for these sequences is currently D N N −1/2 log N ). Nonetheless, there may be other functionals that are as natural and even more efficient.…”

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confidence: 99%

A nonlocal functional promoting low-discrepancy point sets

Steinerberger

2019

Journal of Complexity

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Let X = {x 1 , . . . , x N } ⊂ T d ∼ = [0, 1] d be a set of N points in the d−dimensional torus that we want to arrange as regularly possible. The purpose of this paper is to introduce the energy functionaland to suggest that moving a set X into the direction −∇E(X) may have the effect of increasing regularity of the set in the sense of decreasing discrepancy. We numerically demonstrate the effect for Halton, Hammersley, Kronecker, Niederreiter and Sobol sets. Lattices in d = 2 are critical points of the energy functional, some (possibly all) are strict local minima.2010 Mathematics Subject Classification. 11L03, 42B05, 82C22.

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Dynamically defined sequences with small discrepancy

Abstract: We study the problem of constructing sequences (xn) ∞ n=1 on [0, 1] in such a way that 2010 Mathematics Subject Classification. 11L03, 42B05, 82C22.

Cited by 14 publications

References 15 publications

Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Greedy energy minimization can count in binary: point charges and the van der Corput sequence

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

A nonlocal functional promoting low-discrepancy point sets

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