Fourier-Stieltjes coefficients and asymptotic distribution modulo 1 (Q1070160)

Let R be the set of all finite complex Borel measures \(\mu\) on the unit circle for which \({\hat \mu}\)(n)\(\to 0\) as \(| n| \to \infty\). Furthermore, a Borel subset E of the unit circle is called a W-set if there exists an increasing sequence \((n_ k)\) of positive integers such that for every \(x\in E\) the sequence \((n_ k x)\) has an asymptotic distribution different from the uniform distribution. The main result of this paper says that \(\mu\in R\) if and only if every W-set is a \(\mu\)-null set. This establishes a claim by \textit{Yu. A. Shrejder} [Dokl. Akad. Nauk SSSR, Nov. Ser. 74, 663-664 (1950; Zbl 0039.294)] who did not give a full proof. An analogous result holds if the unit circle is replaced by a general locally compact abelian group. In a note added in proof the author points out that his key Theorem 1 (which says that from any sequence in the unit ball of \(L^ 2(\mu)\) one can extract a subsequence whose Cesàro means converge pointwise \(\mu\)- a.e.) is a standard result of probability theory.