A Remark on Uniformly Distributed Sequences and Riemann Integrability

Bruijn, FA Dick de; Post, KA Karel

doi:10.1016/s1385-7258(68)50016-7

Cited by 15 publications

(5 citation statements)

References 1 publication

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“…Below Theorem 2.7 in Chapter 1 of [17], Krengel mentions that Weyl's theorem is sometimes spelled out for to the class of Riemann-integrable functions. Actually, it was proved by de Bruijn and Post [7] that the function is Riemann-integrable if and only if the convergence is uniform. This also follows from our next proposition.…”

Section: Uniform Convergencementioning

confidence: 99%

On uniform convergence in ergodic theorems for a class of skew product transformations

Brettschneider¹

2011

Discrete &Amp; Continuous Dynamical Systems - A

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Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, µ) in the base and a semigroup of transformations on another probability space (Ω, F, P ) in the fibre. Under suitable mixing conditions for the fibre transformation, we show that the properties ergodicity, weakly mixing, and strongly mixing are passed on from the base transformation to the skew product (with respect to the product measure). We derive ergodic theorems with respect to the skew product on the product space.The main aim of this paper is to establish uniform convergence with respect to the base variable for the series of ergodic averages of a function F on M × Ω along the orbits of such a skew product. Assuming a certain growth condition for the coupling function, a strong mixing condition on the fibre transformation, and continuity and integrability conditions for F, we prove uniform convergence in the base and L p (P )-convergence in the fibre. Under an equicontinuity assumption on F we further show P -almost sure convergence in the fibre. Our work has an application in information theory: It implies convergence of the averages of functions on random fields restricted to parts of stair climbing patterns defined by a direction.

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Section: Uniform Convergencementioning

confidence: 99%

On uniform convergence in ergodic theorems for a class of skew product transformations

Brettschneider¹

2011

Discrete &Amp; Continuous Dynamical Systems - A

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“…This is a considerable problem even for d = 1: The left-hand side only depends on countably many values of f , so any notion of integrability which is preserved under changing f at countably many points is inevitably too weak to keep the conclusion of the theorem intact. There is a clarifying No-Go Theorem: Given any Lebesgue-integrable function f : [0, 1] → R which does not admit a Riemann-integrable representative, there must exist a uniformly distributed sequence (t n ) such that Equation 5.1 fails [dBP68].…”

Section: Equidistribution Argumentsmentioning

confidence: 99%

Torsion homology growth beyond asymptotics

Braunling

2017

Preprint

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We show that (under mild assumptions) the generating function of log homology torsion of a knot exterior has a meromorphic continuation to the entire complex plane. As corollaries, this gives new proofs of (a) the Silver-Williams asymptotic, (b) Fried's theorem on reconstructing the Alexander polynomial (c) Gordon's theorem on periodic homology. Our results generalize to other rank 1 growth phenomena, e.g. Reidemeister-Franz torsion growth for higher-dimensional knots. We also analyze the exceptional cases where the meromorphic continuation does not exist.

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“…Singular integrands cannot be BVHK; they cannot even be Riemann integrable. It is known since [6] and [3] that for any integrand f on [0, 1] d that is not Riemann integrable, there exists a sequence x x x i ∈ [0, 1] d for which the star discrepancy D * n (x x x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Quasi-Monte Carlo for an Integrand with a Singularity Along a Diagonal in the Square

Basu

Owen

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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Quasi-Monte Carlo methods are designed for integrands of bounded variation, and this excludes singular integrands. Several methods are known for integrands that become singular on the boundary of the unit cube [0, 1] d or at isolated possibly unknown points within [0, 1] d . Here we consider functions on the square [0, 1] 2 that may become singular as the point approaches the diagonal line x 1 = x 2 , and we study three quadrature methods. The first method splits the square into two triangles separated by a region around the line of singularity, and applies recently developed triangle QMC rules to the two triangular parts. For functions with a singularity 'no worse than |x 1 − x 2 | −A is' for 0 < A < 1 that method yields an error of O((log(n)/n) (1−A)/2 ). We also consider methods extending the integrand into a region containing the singularity and show that method will not improve upon using two triangles. Finally, we consider transforming the integrand to have a more QMC-friendly singularity along the boundary of the square. This then leads to error rates of O(n −1+ε+A ) when combined with some corner-avoiding Halton points or with randomized QMC but it requires some stronger assumptions on the original singular integrand.

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A Remark on Uniformly Distributed Sequences and Riemann Integrability

Cited by 15 publications

References 1 publication

On uniform convergence in ergodic theorems for a class of skew product transformations

On uniform convergence in ergodic theorems for a class of skew product transformations

Torsion homology growth beyond asymptotics

Quasi-Monte Carlo for an Integrand with a Singularity Along a Diagonal in the Square

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