The law of the iterated logarithm for discrepancies of {θ n x}

Fukuyama, Katusi

doi:10.1007/s10474-007-6201-8

Cited by 58 publications

(87 citation statements)

References 6 publications

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“…Very recently, Fukuyama [6] computed the value of the lim sup for the sequences n k = θ k , where θ > 1, not necessarily integer. Put…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Irregular discrepancy behavior of lacunary series

Aistleitner

2008

Monatsh Math

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In 1975 Philipp showed that for any increasing sequence (n k ) of positive integers satisfying the Hadamard gap condition n k+1 /n k > q > 1, k ≥ 1, the discrepancy D N of (n k x) mod 1 satisfies the law of the iterated logarithmRecently, Fukuyama computed the value of the lim sup for sequences of the form n k = θ k , θ > 1, and in a preceding paper the author gave a Diophantine condition on (n k ) for the value of the limsup to be equal to 1/2, the value obtained in the case of i.i.d. sequences. In this paper we utilize this number-theoretic connection to construct a lacunary sequence (n k ) for which the lim sup in the LIL for the star-discrepancy D * N is not a constant a.e. and is not equal to the lim sup in the LIL for the discrepancy D N .

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“…Very recently, Fukuyama [6] computed the value of the lim sup for the sequences n k = θ k , where θ > 1, not necessarily integer. Put…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Irregular discrepancy behavior of lacunary series

Aistleitner

2008

Monatsh Math

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“…Very recently, Fukuyama [9] determined the lim sup in (1.2) for the sequence n k = θ k , θ > 1 (not necessarily integer). He showed that the lim sup Σ θ equals 1/2 if θ r is irrational for r = 1, 2, .…”

Section: N N X))mentioning

confidence: 99%

“…[15], p. 143) the finiteness of the lim sup in (1.11) follows from Philipp's LIL (1.2) and thus the essential new information provided by Theorem 1.3 is the exact value of the lim sup. It is worth pointing out that replacing L * by L in (1.8) the value of the lim sup can be different from f 2 , as the example n k = 2 k shows; see Fukuyama [9].…”

Section: Theorem 12 Let (N K ) K≥1 Be a Sequence Of Positive Integementioning

confidence: 99%

On the law of the iterated logarithm for the discrepancy of lacunary sequences

Aistleitner

2010

Trans. Amer. Math. Soc.

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Abstract. A classical result of Philipp (1975) states that for any sequence (n k ) k≥1 of integers satisfying the Hadamard gap condition n k+1 /n k ≥ q > 1 (k = 1, 2, . . .), the discrepancy D N of the sequence (n k x) k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e.The value of the lim sup is a long-standing open problem. Recently Fukuyama explicitly calculated the value of the lim sup for n k = θ k , θ > 1, not necessarily integer. We extend Fukuyama's result to a large class of integer sequences (n k ) characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the lim sup is the same as in the Chung-Smirnov LIL for i.i.d. random variables.

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“…For a θ which is a power root of an integer, of a large rational number, or of a ratio of odd integers, the concrete value of Σ θ is evaluated. See [12,14,15,16,17]. For conditions to have an exact law of the iterated logarithm in (3), see [1,5].…”

Section: For Arithmetic Progressions {Kx} With X /mentioning

confidence: 99%

A metric discrepancy result with given speed

2016

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Abstract. It is known that the discrepancy D N {kx} of the se-a.e. for some 0 < Σ θ < ∞ and N ≥ N 0 if ε > 0, but not for ε < 0. In this paper we prove, extending results of Aistleitner-Larcher [6], that for any sufficiently smooth intermediate speed Ψ(N ) between (log N )(log log N ) 1+ε and (N log log N ) 1/2 and for any Σ > 0, there exists a sequence {n k } of positive integers such that N D N {n k x} ≤ (Σ + ε)Ψ(N ) eventually holds a.e. for ε > 0, but not for ε < 0. We also consider a similar problem on the growth of trigonometric sums.

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The law of the iterated logarithm for discrepancies of {θ n x}

Abstract: It is proved that two types of discrepancies of the sequence{θ n x} obey the law of the iterated logarithm with the same constant. The appearing constants are calculated explicitly for most of θ > 1.

Cited by 58 publications

References 6 publications

Irregular discrepancy behavior of lacunary series

Irregular discrepancy behavior of lacunary series

On the law of the iterated logarithm for the discrepancy of lacunary sequences

A metric discrepancy result with given speed

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