Random constructions for translates of non-negative functions

Buczolich, Zoltán and Hanson, B. and Maga, Balázs and Vértesy, Gáspár (2018) Random constructions for translates of non-negative functions. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 468 (1). pp. 491-505. ISSN 0022-247X

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Abstract

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

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