Fourier-Stieltjes coefficients and asymptotic distribution modulo $1$


Let $R$ denote the class of complex Borel measures on the circle $\mathbf{T}$ whose Fourier-Stieltjes coefficients in $\hat\mu(n)$ tend to $0$ as $|n| \to \infty$. Ju. A. Šreǐder has defined a class of subsets of $\mathbf{T}$, called $W$-sets, using the notion of asymptotic distribution. We establish Šreǐder’s unproved claim that a measure $\mu$ lies in $R$ if and oly if $\mu E = 0$ for all $W$-sets $E$. This depends on a remarkable lemma about asymptotic distribution. This lemma is, in turn, a special case of a theorem which allows us to extract from any weakly convergent sequence of functions a subsequence whose Cesàro means converge pointwise almost everywhere.


Russell Lyons