论文标题
几乎到处的一系列非负功能翻译问题的融合问题
Almost everywhere convergence questions of series of translates of non-negative functions
论文作者
论文摘要
该调查文件基于巴黎实际分析的第44夏季研讨会上的演讲。 这一研究是由Haight andWeizsäker的问题引发的，几乎在任何地方都在$ \ sum_ {n = 1}^{\ infty} f（nx）$的一系列融合属性。此问题的更通用的添加剂版本如下： 假设$λ$是一组离散的无限实数。我们说，如果系列$ s（x）= \ sum_ {λ\inλ} f（x+λ）$满足零一个法律，则$λ$是类型1的。这意味着，对于任何不可测量的$ f：{\ mathbb r} \ to [0，+ {\ infty}）$要么收敛set $ c（f，λ）= \ {x：s（x）<+ {x）<+ {\ infty} $ d（f，λ）= \ {x：s（x）=+ {\ infty} \} = {\ mathbb r} $ modulo sem a y house zero的集合。 如果$λ$不是类型1的，我们说$λ$是类型2。 $ 1 $和类型$ 2 $ sets的确切表征仍然不知道。 本文讨论有关此问题的结果的一部分是基于J-P一开始写的几篇联合论文。 Kahane和D. Mauldin，后来与B. Hanson，B。Maga和G.Vértesy一起。 除上述项目的结果外，我们还涵盖了历史背景，其他相关结果和开放问题。
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 Weizsäker concerning almost everywhere convergence properties of series of the form $\sum_{n=1}^{\infty}f(nx)$. 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 $s(x)=\sum_{λ\inΛ}f(x+λ)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {\mathbb R}\to [0,+ {\infty})$ either the convergence set $C(f, Λ)=\{x: s(x)<+ {\infty} \}= {\mathbb R}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, Λ)=\{x: s(x)=+ {\infty} \}= {\mathbb R}$ 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. Vértesy. Apart from results from the above project we also cover historic background, other related results and open questions.
