On a gap series of Mark Kac

• Autorzy: Katusi Fukuyama
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

cocycles; gap theorem; central limit theorem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1999
• Tom: 81
• Numer: 2
• Strony: 157-160

Daty

wydano

1999

otrzymano

1998-07-01

Twórcy

autor

Katusi Fukuyama

• Department of Mathematics, Kobe University, Rokko, Kobe, 657-8501 Japan

Bibliografia

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