On the class of limits of lacunary trigonometric series

Aistleitner, Christoph

doi:10.1007/s10474-010-9218-3

Cited by 3 publications

(3 citation statements)

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“…However, comparing the case of sequences of the form (ξx sn ) n≥1 to the somewhat similar case of lacunary sequences, one sees that it is by no means clear that the (precise) LIL and CLT have to hold for geometric progressions. In the case of lacunary sequences (s n x) n≥1 , the value of the limsup in the LIL for the discrepancy depends on the precise number-theoretic properties of (s n ) n≥1 in a very complicated way, and can even be non-constant (see [2,3,4]). Furthermore, the asymptotic behavior of lacunary sequences can change significantly after a permutation of its terms, see [36,38].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Quantitative uniform distribution results for geometric progressions

Aistleitner

2014

Isr. J. Math.

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By a classical theorem of Koksma the sequence of fractional parts ({x n }) n≥1 is uniformly distributed for almost all values of x > 1. In the present paper we obtain an exact quantitative version of Koksma's theorem, by calculating the precise asymptotic order of the discrepancy of ({ξx sn }) n≥1 for typical values of x (in the sense of Lebesgue measure). Here ξ > 0 is an arbitrary constant, and (s n ) n≥1 can be any increasing sequence of positive integers.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Quantitative uniform distribution results for geometric progressions

Aistleitner

2014

Isr. J. Math.

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“…is "small" (see [5,7,15]), while "irregular" probabilistic behavior as in (4) may occur if the number of solutions of such Diophantine equations is "large" (see [3,8,14]). This also carries over to the LIL for the discrepancy of (n k x) k≥1 : in [4] we showed that if the number of solutions (k 1 , k 2 ), k 1 , k 2 ≤ N , of equations of the form (6) is bounded by O N (log N ) −1−δ for some δ > 0, then hand Fukuyama [12] proved that…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Aistleitner

2012

Trans. Amer. Math. Soc.

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By a classical heuristics, lacunary function systems exhibit many asymptotic properties which are typical for systems of independent random variables. For example, for a large class of functions f the system (f (n k x)) k≥1 , where (n k) k≥1 is a lacunary sequence of integers, satisfies a law of the iterated logarithm (LIL) of the form (1) c 1 ≤ lim sup N →∞ N k=1 f (n k x) √ 2N log log N ≤ c 2 a.e., where c 1 , c 2 are appropriate positive constants. In a previous paper we gave a criterion, formulated in terms of the number of solutions of certain linear Diophantine equations, which guarantees that the value of the lim sup in (1) equals the L 2-norm of f for a.e. x, which is exactly what one would also expect in the case of i.i.d. random variables. This result can be used to prove a precise LIL for the discrepancy of (n k x) k≥1 , which corresponds to the Chung-Smirnov LIL for the Kolmogorov-Smirnov-statistic of i.i.d. random variables. In the present paper we give a full solution of the problem in the case of "stationary" Diophantine behavior, by this means providing a unifying explanation of the aforementioned "regular" LIL behavior and the "irregular" LIL behavior which has been observed by Kac, Erdős, Fortet and others.

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“…for lacunary (n k ) (cf. [2,3,4]). The (nonconstant) limsup in the case n k = 2 k − 1 was determined by Fukuyama [36].…”

Section: Lacunary Seriesmentioning

confidence: 99%