Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8959552 | Journal of Mathematical Analysis and Applications | 2018 | 15 Pages |
Abstract
Suppose Î is a discrete infinite set of nonnegative real numbers. We say that Î is type 2 if the series s(x)=âλâÎf(x+λ) does not satisfy a zero-one law. This means that we can find a non-negative measurable “witness function” f:Râ[0,+â) such that both the convergence set C(f,Î)={x:s(x)<+â} and its complement the divergence set D(f,Î)={x:s(x)=+â} are of positive Lebesgue measure. If Î is not type 2 we say that Î is type 1. The main result of our paper answers a question raised by Z. Buczolich, J-P. Kahane, and D. Mauldin. By a random construction we show that one can always choose a witness function which is the characteristic function of a measurable set. We also consider the effect on the type of a set Î if we randomly delete its elements. Motivated by results concerning weighted sums âcnf(nx) and the Khinchin conjecture, we also discuss some results about weighted sumsân=1âcnf(x+λn).
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Zoltán Buczolich, Bruce Hanson, Balázs Maga, Gáspár Vértesy,