On Sidon Sets and related topics in additive number theory.
Item
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Title
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On Sidon Sets and related topics in additive number theory.
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Identifier
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AAI9605601
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identifier
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9605601
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Creator
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Helm, Martin.
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Contributor
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Adviser: Melvyn B. Nathanson
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Date
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1995
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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A non-empty subset A of N is called a B{dollar}\sb{lcub}r{rcub}{dollar}-sequence if every n {dollar}\in{dollar} N has at most one representation of the form {dollar}n = a\sb1 +\cdot\cdot\cdot + a\sb{lcub}r{rcub}{dollar} with {dollar}a\sb{lcub}i{rcub}\in A{dollar} and {dollar}a\sb1\leq\cdot\cdot\cdot\leq a\sb{lcub}r{rcub}.{dollar}.;In the special case r = 2, {dollar}B\sb2{dollar}-sequences are also called Sidon Sets. This work is devoted to the study of {dollar}B\sb{lcub}r{rcub}{dollar}-sequences, additive bases and related topics in additive number theory.;Chapter 1 investigates an old and attractive conjecture due to P. Erdos that asserts that the counting function {dollar}A(n):= \Sigma\sb{lcub}a\in A,1\leq a \leq n{rcub}{dollar}1 of a {dollar}B\sb{lcub}r{rcub}{dollar}-sequence A satisfies lim inf{dollar}\sb{lcub}n\rightarrow\infty{rcub}A(n)\sp{lcub}-1/r{rcub}{dollar} = 0.;In particular, Section 1.3.1. provides a detailed exposition of a proof of Erdos' conjecture in the even case r = 2k. Furthermore 1.3.2. will be concerned with the improvement of recent results of Chen on {dollar}B\sb{lcub}2k{rcub}{dollar}-sequences. Chapter 1.4. discusses the case of {dollar}B\sb{lcub}2k+1{rcub}{dollar}-sequences and is primarily concentrated on {dollar}B\sb3{dollar}-sequences.;We prove that no sequence of pseudo-cubes i.e, a sequence A whose counting function satisfies {dollar}A(n)\sim\alpha\ n\sp{lcub}1/3{rcub}{dollar} for some {dollar}\alpha{dollar} is a {dollar}B\sb3{dollar}-sequence. Section 1.4.1. establishes various results on the distribution of the elements of a given {dollar}B\sb3{dollar}-sequence.;Another interesting conjecture of P. Erdos states that there exists a {dollar}B\sb3{dollar}-sequence A that satisfies lim sup{dollar}\sb{lcub}n\rightarrow\infty{rcub} A(n)\ n\sp{lcub}-1/3{rcub}{dollar} = 1. Using a result of Erdos on sum-free sets of integers we construct an infinite sequence of natural numbers that is not "too far" away from being a {dollar}B\sb3{dollar}-sequence and that at the same time satisfies lim sup{dollar}\sb{lcub}n\rightarrow\infty{rcub} A(n)\ n\sp{lcub}-1/3{rcub}\geq{dollar} 1.;Chapter 2 is intended to present some recent results on the Erdos-Turan conjecture. The Erdos-Turan conjecture suggests that there exists no asymptotic basis A of order 2 of N, such that the number of representations of natural numbers n as n = a + b with {dollar}a, b\in A{dollar} is bounded. Section 2.1 proves by means of an explicit construction that a specific result of Erdos that is closely related to a potential proof of the Erdos-Turan conjecture is sharp with respect to magnitude.;Chapter 3 is devoted to the application of probabilistic tools in additive number theory.;In Section 3.1. some basic facts about the probabilistic method are compiled. Section 3.2. indicates how these techniques are used to generalize a well-known result of Erdos on asymptotic bases of order 2.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
