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A General Theory of Asymptotic Density + Natimai Library Bibliothbque nationale CANADIAN THESES of Canada du Canada ON MICRWICHE I I UN IVERSITY/UN/VERSIT# *FRP,~LIL DEGREE F& WHICH THESIS WAS mES€NTED/ GRADE POUR LEQUEL CETTE THESE NTP~~SENT~E 3 Permission is hmy-gr'mted to the NATIONAL LIBRARY OF ' , L'autwisation est,' per la prdsente, accord& b la, BIBLIOTH~- CANADA to microfilm this thesis and to led or sell copies QUE NATIONALE DU CANADA de micrdilmer cette these et . < t of the film. , , , de prefer w de vendre des exemplaires du film. 1 \ . 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If pages are missing, contact the university which S'il manque des pages, veuillez communiquer avec granted the degree. I'universite qui a confere le grade. - - - - - - - - --Lp - Some pages may'have indistinct print especially if La qualite d'impression de certaines pages peut the original pages were typed with a poor typewriter laisser a dhirer, surtout si les pages originales ont Bt6 ribbon or if the university sent us a poor photocopy,, ' dactylographiees a I'aided'un rwban use ou si I'universite nous a fait parvenir une photocopie de mauvaise qualite. Previously copyrighted materials (journal articles, . Les documents qui font deja I'objet d'un droit d'au- published tests, etc.)are ndfilmed. -- teur (articles de revue, examens publies, etk.) ne sont pas microf ilmes.- k ( Reproduction in full or in part of this film is governtd La reproduction, mgme partielle, de ce microfilm esf by the Canadian Copyright Act, R.S.C. 1970, c. C-30. soumi& 2 19 ~oicanadienne sur le droit d'auteur, SPC Please read the authorization forms which accompany 1970,' c. C-30.Veuillez prendre connaissance des for- this thesis. mules d'autorisation qui acc~mpa~nent-cettethese. , THIS DISSERTATION LA TH~EA ETE HAS BEEN MICROFILMED MICROFILM~ETELLE QUE EXACTLY AS RECEIVED NOUS L'AVONS RECUE J A GENERAL THEORY OF ASYMPTOTIC DENSITY + / . r - Dan .Sonnens+ein ; r 1 B.A., University of British Columbia, 1971 a \ THE ReQUIREPa3TS FOR THE DEGREE OF t MASTER oI? SCIE& ' ? in the Departraent . Mathematics 6 . - @ DAN JACOB SONN~NSCHEIN 19 7 8 SIMON F'RASER UNIVERSITY f - a , March 1978 All rights reserved. his thesis may not be reproduced ih hole or in part, by photocopy or other means, without permission of the author. 5 - APPROVAL Name: Dan Jacob Sonnenschein Degree; Master of Science / ~itieof Thesis: A General Theory of Asymptotic ksit~ Chairperson: Alistair,H. Lachlan 7 Allen R. Freeclman ' Senior Supervisor,'* John J. ember ' ,:, , , .. I Norman 1R. Reilly.' G4 A. Cecil Graham I .. External Examiner J &ch fb, 197-8 , Date-fipproved : PARTIAL COPYRIGHT LICENSE e . I hereby grant to Simon Fraser University the right to lend my .thesis. or dissertation (the title of which is shown below) to users - i ,. of the Simon Fraser*University Library, and to make partial or single *r P copies only for such users or ,in response to a request from the library ~f an)l dhpr mimsi% or -- > - behalf or for one of its users. I fu,rther agree that penission far f mu1 tiple copying of this thesis for scholarly purposes may. be granted - - - by me or the Dean of Graduate Studies. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my wrikten permission. 1 G3 - Ti tle of Thesis/Dissertation : . " ', . - < Author : -, I ' 4 .- - - - (signatur'e) . r . (date) . C / me- usual low& d'srymptotic aendity is. generalized by defining , ' an abstract density as a set 'functiongatisfyiW certain axioms. A ' /. generalized upper densie and natural density are defined, and elemeniary .. , .d i * < '1 properties of dengity, upper 'dellsity and natural density &e pmved. .a This abstract density theory is -iq pakt analogous to measure theory, A-. I GCePTthat countable. atfditivity for natural. density does not hold. Various asyrnptoticqhensitiesf -t including the usual one are shown - of suxmnability. ~ecessar~and sufficient conditions on a matrix in ..A order for its associated set function to be an asymptotic * -. 1- d given. An approximatiion to Gountable additivity for natural -density is B. shown to hold for the class of densities associated*with regular matrix. 3 =. he order relations among particular densities are discussed, t B ant3 a.natural density. which is an extension of the natural ordinary B density is presented.- Some applications to summability theory and number ' theory aTe given, and some unLsolvedproblems are.stated. .- '. I? = DEDICATION 6 I would like to express my appreciation to Dr. A. R. Preet3ma.n for the main ideas *and most of -the results of this thesis, and to - Dr. J. J. Sember for- stimulating discussion on sae of these topic& would also like to thank Dr. A. A. Lachlan for arranging th9 research P C assistanstshipmat provided me with extra time to concentrate on J completion of the thesis. Finally, I wuld ljke to thank Mrs. Nancy Hood for the typing of the entire work. * - .- r 't . .- TABLE OF CONTENTS * * * Title Page i * Approval Dedication, , I Acknowledgements v 'Table of Contents 2 Axioms and,Elementary Properties ,' $3 Matrix Methods and the Additivity Property $4 ~belDensity $5 Uniform Density - $6 Logarithmic Density Z .. $7 Conclusion Bibliography J 6 s ~etI denote the set of positive integers, and let 2' denote the class of subsets of I. For A C 2', define the counting + function Ah) as' the number of elements in A fl 1, , n. The . , I;, usual (or ordinary) lower asymptotic density 6 is defined by ', 4 * - .- - 6ik) - n - - , afla che usual upper asymptotic density 6 is defined by 6 (A) = lim sup -A(n) . If the ordinary lower and *upper 3 n n - 4 densities of A are equal, A is said td have natural ordinafy density -- -- -- asymptotic and banslation invariant properees;.which are definedA Chapter 2. There two axiom systems are given, defining an abstract - % -totic density and an abstract translyiion invariant densic\ both of which are generalizations of the usual' lower asymptotic densiey. It * .. , I is shown that a translation invariant density is an as&tot& densi'ty, 2. 0 54that an ashtotic density is not necessarily tralation inkrjant. The term density is used , - c An upper density is defined in terms of the density, and a '1 generalized naGural density is defined on the class of sets with equal #. density and upper density. Various elementary properties of density 1 C - - - / upper density, and natural density are proved in Chapter 2. Of , r a -- -- patti- =~rt~c5is~hZEwhile~~n~uraldensityis finitely- , r additive, in contrast to a measure it is not countably additive.. 9 Buck 111 has defined in a different way a finitely additive I .* . - C density fdnction, which has other properties of measure as well:, He t 0 'alsq.-defines a dass of "l*it densities," based on the observation that ' , the natubl ordinary density of a set, when it exists, is the Cesaro ' lipit of the characteristic s&ehce of the Set. (The usual lower and s d 9 upper densities of 'a set are the 1g inf and lim sup- 'respectively, . 1 d r --or the msaro transfomLof the characteristic sequence of the se-t,) ' This observation suggests the possibili'ty of applying .otIierxsurmnability . " - - method; in the sa& way to obtai-nb&herclimitdensities. ThC. cogme_ctJi_on- - 7 between density and re*gular matrix methois of s-ility ij studied.in /' % . , I I " Chapter 3. There we give n&cessary*and sufficient condikions Qh a ' ' - 4 retjular matrix in order for its associated setpfunc.&oyu to be an . - . i -CI e - 1'1 C . a-totic *nbty. Another main result of. fhis ch~rpter.is that an + /I I * . 4 i - .. ?I - - \approximation to countable additivity for natural denslty, c$lled the 4. 1(4 . T* *. -* n addisivity property, class of densities associated with ' . 2 .. .. 5 regular lnatr~xmethods of surmnability: -. $ 7 - / J. The densitb st6died in Chapter 4 :is obtained from the Abel . .is C * ea * - I -- sunrmability method, a regular 'bsemi-continuous matrix method" of .+ . , -0 . 'a b - ' \* , X '- -'.- 3 ,. --*. s&mkbility stronger than Cesaro sumability . well-knob results &e 4' 4' - * stated which show that natural Abel density is equivalent to natkal 3 - density,. .and thus that -1 denky has the additivity property. - - --- - - -- -- - -- The uniform ''amd ity of chip&r 4 is associated with the -- - \ -- - - -- t summability method of "almost convergence," which is not a regular matrix I I. method. An ex*le is given to show that uniform densit$ does not have i the additivity property. Natural ordinary density is shown to be a ' 2 Ii a-'i proper extension of natura1,uniform density. \ i f The logarithmic density of Chapter 5 is'&ther density * abtained frcm a regul-kr matrix method of sumnability, and so it has the * additivity property. Known results are stated which show that natural / * / logarithmic density G a proper extension of natural or&& density.
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