On series of translates of positive functions. II
This paper is a continuation of the earlier paper [\textit{Z. Buczolich, J.-P. Kahane} and \textit{R. D. Mauldin}, Acta Math. Hung. 93, 171-188 (2001; Zbl 1006.28005)]. It gives a detailed study of the set of convergence of series of the form \(\sum_{\lambda \in \Lambda} f(x+\lambda)\) for a nonnegative measurable function \(f.\) From the authors' summary: ``Given a sequence of natural numbers \(n_1 < n_2 < \ldots \) we are interested in sets \(\Lambda\) of the form \(\bigcup^ {(x)}_{k=1} \alpha ^ k \mathbb{Z} \cap [n_k, n_{k+1}),\) where \(0<\alpha < 1.\) In case \(\alpha = 1/q,\) where \(q>1\) is an integer, there is a zero-one law showing that for every measurable \(f : \mathbb{R} \rightarrow \mathbb{R}^ +\) the above sum either converges almost everywhere or diverges almost everywhere. However, for any other value of \(\alpha \in (0,1)\) there is no such zero-one law.'' The main work is in constructing explicitely measurable sets \(A \subset \mathbb{R}\) such that \(f=1_A\) serves as counterexample.
scientific article; zbMATH DE number 1766473