Uniform Distribution, Behrend Sequences, and Some Spaces of Arithmetic Functions
Hill, Christopher Brooks
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https://hdl.handle.net/2142/87000
Description
Title
Uniform Distribution, Behrend Sequences, and Some Spaces of Arithmetic Functions
Author(s)
Hill, Christopher Brooks
Issue Date
2000
Doctoral Committee Chair(s)
Hildebrand, A.J.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
This work is comprised of three investigations, each relating to the concept of a density test set. We define a density test set to be a set S of positive integers greater than 1 so that if a set A of positive integers contains the same fraction of all positive multiples of s for each s in S, then A contains the fraction s of all positive integers. The first investigation relates density test sets to uniform distribution modulo 1, leading to conditions under which the uniform distribution of certain subsequences of a sequence of real numbers in [0,1) implies the uniform distribution of the underlying sequence. These conditions strengthen a result of G. Myerson and A. D. Pollington, and yield a characterization of density test sets with prime elements. We conjecture that density test sets are precisely the Behrend sets. In the second investigation, Behrend sets whose elements are products of exactly two prime factors are characterized. Our result differs from the characterization obtained by I. Z. Ruzsa and G. Tenenbaum in 1996, and it verifies the above conjecture for density test sets whose elements are products of exactly two prime factors. The third investigation focuses on a transform of arithmetic functions designed so that a density test set can be defined in terms of the action of the transform on characteristic functions. The domain of the transform is the set of functions that possess Ramanujan expansions. We establish several results on the action of the transform on certain spaces of arithmetic functions considered in depth by W. Schwarz and J. Spilker, in particular the space of even functions and its uniform closure.
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