On Sidon Sets and related topics in additive number theory

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 = asb1 +cdotcdotcdot + asb{lcub}r{rcub}{dollar} with {dollar}asb{lcub}i{rcub}in A{dollar} and {dollar}asb1leqcdotcdotcdotleq asb{lcub}r{rcub}.{dollar}; In the special case r = 2, {dollar}Bsb2{dollar}-sequences are also called Sidon Sets. This work is devoted to the study of {dollar}Bsb{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):= Sigmasb{lcub}ain A,1leq a leq n{rcub}{dollar}1 of a {dollar}Bsb{lcub}r{rcub}{dollar}-sequence A satisfies lim inf{dollar}sb{lcub}nrightarrowinfty{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}Bsb{lcub}2k{rcub}{dollar}-sequences. Chapter 1.4. discusses the case of {dollar}Bsb{lcub}2k+1{rcub}{dollar}-sequences and is primarily concentrated on {dollar}Bsb3{dollar}-sequences.; We prove that no sequence of pseudo-cubes i.e, a sequence A whose counting function satisfies {dollar}A(n)simalpha nsp{lcub}1/3{rcub}{dollar} for some {dollar}alpha{dollar} is a {dollar}Bsb3{dollar}-sequence. Section 1.4.1. establishes various results on the distribution of the elements of a given {dollar}Bsb3{dollar}-sequence.; Another interesting conjecture of P. Erdos states that there exists a {dollar}Bsb3{dollar}-sequence A that satisfies lim sup{dollar}sb{lcub}nrightarrowinfty{rcub} A(n) nsp{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}Bsb3{dollar}-sequence and that at the same time satisfies lim sup{dollar}sb{lcub}nrightarrowinfty{rcub} A(n) nsp{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, bin 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.