## On a gap series of Mark Kac

### Volume 81 / 1999

Colloquium Mathematicum 81 (1999), 157-160
DOI: 10.4064/cm-81-2-157-160

#### 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.