The measure of non-normal sets (Q1071053)

A number \(x\in {\mathbb{T}}={\mathbb{R}}/{\mathbb{Z}}\) is said to be normal with respect to the base \(q\geq 2\) if each digit \(a\in \{0,1,...,q-1\}\) occurs in the q-adic development of x with the same frequency \((=1/q)\). It is well- known that in the sense of Lebesgue measure \(\lambda\) almost all real numbers are normal. Hence \(\lambda (M)=0\), where M denotes the set of all non-normal numbers. In this paper the negative answer is given to the question of J. P. Kahane and R. Salem whether \(\mu (M)=0\) for each Borel measure \(\mu\) whose Fourier-Stieltjes coefficients vanish at infinity. For the Borel measure \(\mu\) we put \[ {\hat \mu}(n)=\int_{T}e^{-2\pi int} d\mu (t) \] In the paper certain ''best possible'' conditions on \({\hat \mu}\) are given in order that \(\mu\)-almost all real numbers are normal.