Some results in additive number theory.
Item
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Title
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Some results in additive number theory.
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Identifier
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AAI9029946
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identifier
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9029946
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Creator
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Jia, Xing-De.
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Contributor
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Adviser: Melvyn B. Nathanson
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Date
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1990
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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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The thesis is devoted to the study of bases in additive number theory. It contains four chapters.;Chapter One investigates the order of subsets of asymptotic bases. Let {dollar}g(A){dollar} denote the smallest integer {dollar}h{dollar} such that the set {dollar}A{dollar} is an asymptotic basis of order {dollar}h{dollar}. Some estimates are proved for the extremal function{dollar}{dollar}G\sb{lcub}k{rcub}(h)={lcub}\max\limits\sb{lcub}g(A)\le h{rcub}{rcub}\ {lcub}\max\limits\sb{lcub}\vert F\vert=k\atop g(A\\ F)<\infty{rcub}{rcub}\ g(A\\ F),{dollar}{dollar}including{dollar}{dollar}G\sb{lcub}k{rcub}(h)\ge(k+1)\left({lcub}k + 1\over k+2{rcub}\right)\sp{lcub}k{rcub}\left({lcub}h\over k+1{rcub}\right)\sp{lcub}k+1{rcub} + O(h\sp{lcub}k{rcub}){dollar}{dollar}as {dollar}h{dollar} tends to infinity for any fixed {dollar}k{dollar}. It is also proved that {dollar}G\sb{lcub}k{rcub}(h){dollar} has order of magnitude {dollar}k\sp{lcub}h-1{rcub}{dollar} as {dollar}k{dollar} tends to infinity for any fixed {dollar}h{dollar}. An interesting connection between this problem and the theory of extremal bases in the postage stamp problem is also proved in this chapter.;Chapter Two describes a simple and explicit construction of minimal asymptotic bases of order {dollar}h{dollar} for every {dollar}h \ge 2{dollar} by using either powers of 2 or {dollar}g{dollar}-adic representations of integers. It is also proved in this chapter that there exist minimal bases for commutative monoids, which generalizes some results in additive number theory concerning minimal bases.;In Chapter Three, it is proved that if {dollar}\Phi = \{lcub}S\sb1,S\sb2,\...,S\sb{lcub}s{rcub}\{rcub}{dollar} and {dollar}\Psi = \{lcub}T\sb1,T\sb2,\...,T\sb{lcub}t{rcub}\{rcub}{dollar} are two families of nonempty, pairwise disjoint sets such that {dollar}\vert S\sb{lcub}i{rcub}\vert \le h, \vert T\sb{lcub}j{rcub}\vert \le k (h \ge 2{dollar} and {dollar}k \ge 1){dollar} and {dollar}S\sb{lcub}i{rcub} \not\subseteq T\sb{lcub}j{rcub}{dollar} for all {dollar}i{dollar} and {dollar}j{dollar}, then{dollar}{dollar}N(\Phi,\Psi)\le h\sp{lcub}s{rcub}\left(1-{lcub}h-r\over h\sp{lcub}q+1{rcub}{rcub}\right)\sp{lcub}t{rcub},{dollar}{dollar}where {dollar}k = q(h - 1) + r{dollar} with 0 {dollar}\le r \le h - 2,{dollar} and {dollar}N(\Phi,\Psi){dollar} is the number of sets {dollar}X{dollar} such that {dollar}X{dollar} is a minimal system of representatives for {dollar}\Phi{dollar} and {dollar}X{dollar} is simultaneously a system of representatives for {dollar}\Psi{dollar}. This was a conjecture of Nathanson.;Chapter Four considers the existence of thin bases for finite groups. It is also proved in this chapter that bases with given number of representations exist for certain infinite abelian groups. (Abstract shortened with permission of author.).
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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.
