• Corpus ID: 252531405

Almost everywhere convergence questions of series of translates of non-negative functions

@inproceedings{Buczolich2022AlmostEC,
  title={Almost everywhere convergence questions of series of translates of non-negative functions},
  author={Zolt{\'a}n Buczolich},
  year={2022}
}

• Z. Buczolich
• Published 24 September 2022
• Mathematics

This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris. This line of research was initiated by a question of Haight and Weizs¨aker concerning almost everywhere convergence properties of series of the form P ∞ n =1 f ( nx ). A more general, additive version of this problem is the following: Suppose Λ is a discrete infinite set of nonnegative real numbers. We say that Λ is of type 1 if the series s ( x ) = P λ ∈ Λ f ( x + λ ) satisfies a zero-one law. This means that for any non-negative measurable f : R → [0 , + ∞ ) either the convergence set C ( f, Λ) = { x : s ( x ) < + ∞} = R modulo sets of Lebesgue zero, or its complement the divergence set D ( f, Λ) = { x : s ( x ) = + ∞} = R modulo sets of measure zero. If Λ is not of type 1 we say that Λ is of type 2. The exact characterization of type 1 and type 2 sets is still not known. The part of the paper discussing results concerning this question is based on several joint papers written at the beginning with J-P. Kahane and D. Mauldin, later with B. Hanson, B. Maga and G. V´ertesy. Apart from results from the above project we also cover historic background, other related results and open questions.