Irregularities of continuous distributions

Drmota, Michael

doi:10.5802/aif.1175

Cited by 7 publications

(2 citation statements)

References 7 publications

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“…for all x, almost all α and all S ∈ B. Moreover, by a careful consideration of the constants involved in the proof of Theorem 1.3, one can verify that the O-constant will depend only on α, and not on the choice of rectangle S. As a consequence, we obtain the following result, previously shown by Drmota [4] (see also [5]). As clarified by the example in Figure 1, an analogous result does not hold if B is the class of all rectangles.…”

Section: (B)supporting

confidence: 67%

Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$

Grepstad

Larcher

2016

Acta Arith.

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We study sets of bounded remainder for the twodimensional continuous irrational rotation ({x 1 + t}, {x 2 + tα}) t 0 in the unit square. In particular, we show that for almost all α and every starting point (x 1 , x 2 ), every polygon S with no edge of slope α is a set of bounded remainder. Moreover, every convex set S whose boundary is twice continuously differentiable with positive curvature at every point is a bounded remainder set for almost all α and every starting point (x 1 , x 2 ). Finally we show that these assertions are, in some sense, best possible. Date

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Section: (B)supporting

confidence: 67%

Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$

Grepstad

Larcher

2016

Acta Arith.

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“…For the sake of simplicity, let us only consider directions of the form α = (α 1 , 1). Drmota [3] showed that if there exists a constant η < 2 such that the inequality nα 1 < |n| −η has finitely many integer solutions n ∈ Z, then for any axis parallel box R ⊆ [0, 1] 2 we have ∆ T (0, α, R) = O(1). In fact, the implied constant depends only on α, which means that by letting R denote the family of axis parallel boxes in [0, 1] 2 , the discrepancy sup R∈R |∆ T (0, α, R)| is also O(1).…”

Section: Introductionmentioning

confidence: 99%

Bounded error uniformity of the linear flow on the torus

Borda

2018

Monatsh Math

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A linear flow on the torus R d /Z d is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over Q. In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne-Ehrenfest type theorem for the mod 1 discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.

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Super-Uniformity of The Typical Billiard Path

Beck

2010

Bolyai Society Mathematical Studies

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Irregularities of continuous distributions

Cited by 7 publications

References 7 publications

Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$

Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$

Bounded error uniformity of the linear flow on the torus

Super-Uniformity of The Typical Billiard Path

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