Random constructions for translates of non-negative functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:468

Article

Buczolich, Zoltan¹  Hanson, Bruce²  Maga, Balazs¹  Vertesy, Gaspar¹ 

[1] Eotvos Lorand Univ, Dept Anal, Pezmany Peter Setany 1-C, H-1117 Budapest, Hungary

[2] St Olaf Coll, Dept Math Stat & Comp Sci, Northfield, MN 55057 USA

关键词: Almost everywhere convergence;    Asymptotically dense;    Borel-Cantelli lemma;    Laws of large numbers;    Zero-one laws;   

DOI  :  10.1016/j.jmaa.2018.08.030

来源: Elsevier

【 摘 要 】

Suppose A is a discrete infinite set of nonnegative real numbers. We say that A is type 2 if the series s(x) = Sigma lambda Lambda f (x + lambda) does not satisfy a zero-one law. This means that we can find a non-negative measurable witness function f : R -> [0,+ infinity) such that both the convergence set C(f, Lambda) ={x : s(x) < + infinity} and its complement the divergence set D (f, Lambda) = {x : s(x) = +infinity} are of positive Lebesgue measure. If Lambda is not type 2 we say that A 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 A if we randomly delete its elements. Motivated by results concerning weighted sums Sigma c(n)f(nx)and the Khinchin conjecture, we also discuss some results about weighted sums Sigma(n=1) (infinity)c(n)f(x + lambda(n).) (c) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free