On series of translates of positives functions

Given f : R --> R+ and Lambda discrete in R+ we denote by C(f, Lambda), res p. D(f, Lambda), the x-set where the series Sigma f(x + lambda) (lambda is an element of Lambda) converges, resp. diverges. The sets Lambda break into two types. Type 1 consists of Lambda such that the Lebesgue measure of eit her C(f, Lambda) or D(f, Lambda) vanishes whatever f measurable, and type 2 consists of all the other Lambda. Buczolich and Mauldin proved that {log n } (n = 1, 2,...) is of type 2. Type 2 is generic, type 1 is rare, and we gi ve examples of both cases (Theorems 1, 2, 3). (C) 1999 Academie des Science s / Editions scientifiques et medicales Elsevier SAS.