# On a gap series of Mark Kac

Colloquium Mathematicae (1999)

- Volume: 81, Issue: 2, page 157-160
- ISSN: 0010-1354

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topFukuyama, Katusi. "On a gap series of Mark Kac." Colloquium Mathematicae 81.2 (1999): 157-160. <http://eudml.org/doc/210733>.

@article{Fukuyama1999,

abstract = {Mark Kac gave an example of a function f on the unit interval such that f cannot be written as f(t)=g(2t)-g(t) with an integrable function g, but the limiting variance of $n^\{-1/2\}\sum _\{k=0\}^\{n-1\} f(2^kt)$ vanishes. It is proved that there is no measurable g such that f(t)=g(2t)-g(t). It is also proved that there is a non-measurable g which satisfies this equality.},

author = {Fukuyama, Katusi},

journal = {Colloquium Mathematicae},

keywords = {cocycles; gap theorem; central limit theorem; central limit theorem for dyadic transformations},

language = {eng},

number = {2},

pages = {157-160},

title = {On a gap series of Mark Kac},

url = {http://eudml.org/doc/210733},

volume = {81},

year = {1999},

}

TY - JOUR

AU - Fukuyama, Katusi

TI - On a gap series of Mark Kac

JO - Colloquium Mathematicae

PY - 1999

VL - 81

IS - 2

SP - 157

EP - 160

AB - Mark Kac gave an example of a function f on the unit interval such that f cannot be written as f(t)=g(2t)-g(t) with an integrable function g, but the limiting variance of $n^{-1/2}\sum _{k=0}^{n-1} f(2^kt)$ vanishes. It is proved that there is no measurable g such that f(t)=g(2t)-g(t). It is also proved that there is a non-measurable g which satisfies this equality.

LA - eng

KW - cocycles; gap theorem; central limit theorem; central limit theorem for dyadic transformations

UR - http://eudml.org/doc/210733

ER -

## References

top- [1] R. Fortet, Sur une suite également répartie, Studia Math. 9 (1940), 54-69. Zbl66.1298.01
- [2] K. Fukuyama, The central limit theorem for Riesz-Raikov sums, Probab. Theory Related Fields 100 (1994), 57-75. Zbl0803.60021
- [3] M. Kac, On the distribution of values of sums of the type $\sum f\left({2}^{k}t\right)$, Ann. of Math. 47 (1946), 33-49. Zbl0063.03091
- [4] R. Salem and A. Zygmund, On lacunary trigonometric series II, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 54-62. Zbl0029.35601

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