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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.
9 In the conclusion, we indicate some of the uses of the concept.
of density in suvgnability theory and number theory-, and state-some of -- the unsolved probl& o;f this general theory of asymptotic density.
me notation and+some conventions that will be used are as / E follows . ~ekerencesto a set A or "all An will mean a set A € 2I,
or "all A € 2'. Unless G'kherwise indicated, lqwercase English letters
refer to positive integers, so that for example, "all i" means "all
i € I." The set-theoretic differeke (a C A 1 a X, B) is denoted by
A \ B, and the complement I \ A is denoted by A. Proper containment
is denoted by c,' and containment by C. ina all^, the order of 'the -. terms in many infinite series is rearranged without explicitly mentioning the absolute convergence. that allows it to be done. * . In this chapter, two sets of axiosts for an abstract density
t are given. These axiaa sys'tesps define an asyarptotic density and 4
translation invariant density, both of which are gener&lizations of ,the, I % .! usual lower asymptotic density. It will be shm that a translation f
I in9ariant density is an asymptotic dbnsity, but that an asymptotic O , density is not-necessarily translation invariant. Unless otherwise inaicatt&, me- word -densityn is used to refer to bth types of density.
An abstract upper density is defined in terms of the density, and a set is said to have natural density if its density and upper density are equal.$ There is a partial analogy of density, upper density, and natural density to inner measure, outer measure, and measure . respectively. The analogy is not complete because countable additivity of a measure d&s not carry over to countable additivity. of a natural . I-..*,., density. However, an approximation to countable additivity will be shown toehold for some natural densities. g We begin by givin* sane definitions which are used in the
formulation of the axioms. ,*- l
~efinition2.2. Set A is asymptotic to set B, written
A - B, means that the symmetric difference A A B' is finite.
The relation - is easily seen to be reflexive and syraatetric. 2 1 ff;lstransitive as well, since if bath A A B and B h C are finite, I C
relation, and so we aaay refer to asymptotic sets as equivalent sets. A i e - -,-.- useful charact&i%ation of equivalent sets is given by the following
Lemma 2.1. .A B if and only if there exists an N such'
that A \ 1 . . . N = B \ 1, . . . N or A U {l, . . . ,N} =
B u {1, . . . ,El}. -
Proof. If A- B, so that A A B 'is finite, there is &I
initial segment 1, . . . N containing the finitely many integers
that are in exactly one of A or B. Then.an integer greater than N
is in neither set or in both sets, so that A \ {I, . . . ,N} =
B \ 1 . . . N and A U 1, . . . N = B u 1 . . . . If
either of the latter conditions holds* for some N, then there are at most N integers in exactly one of A or B, so that A A B is finite and A - B.
!&ro systems of axioms for a function d. on 2* are now given, defining the two abstract densities. The axioms of each set will be sham to be independent. Tfre first set ist
- - - - 1. for all A, O 5 d(A) 5 1;
3. if A-B, then dW = d(B1; - -- 5. for all A, B, d(A) + d(B) I1+-d(A fl B).
The second set of axioms is obtained from the first by replacing Axiom 3
With:
3'. for all A, d(A + 1) = d(A).
'' density if it satisfies the firs; ~6%of ax"B$ihs. S ' rL
Definition 2.4. A function d on 2' *iscalled a transhtion
invariant density if it satisfies the second set of axians.
Theorem 2.1. The axicrms in each of the above systems are
independent. -
- &ere 6 is the usual lower density. Then d (I) = 1 xiam am 2) ; -. - d(~+1) = ~s(A+1) - 1 = 26@) -I=d(~) for all A (Axiom 3');
and if A-B, then d(A) = 26h) - 1 = 26(B) - 1 = d(B) (Axiom -.3).\ . Also, if A fI B = 4, then d@) + d(B) = 2(6(~)+ 6(~))- 25 26(~UB) - -
2 < 26(A U B) - 1 = d(A U B) (Axiam 4); and for all A, B,
1 + d(~nB) (Axiom 5). But for 6(A) < d(A) < 0, so Axian 1 does 7-'------
not hold, and is therefore independent- - in.- both -sets. -- -
The independence of Axiom 2 is shown by defining d on 2I 1 by d(A) = 7 6(A). It follows easily that all axioms except Axiom 2 hold. I 'For Axioms 3 and 3', consider the function d on 2 1 i . defined by d01) = 2 - Then clearly Axioms L and 2 hold. If 2a aEA A n B = 4, then d (A) e so that Axiom 4 holds [with equality). Now using this axiom and Axian 1,
1 1 1 2z -= C -l+ Z -l+ 2 A= Z -+ z -5 2" nhAflB 2n -n€AdB 2n nfAflB 2n n€Afl~2n ~CAUB 2n n€AflB 1 + d (A fl B) , so Axim 5 is satisfied. But for A = 113, A + 1 -. A
and d(A + 1) # d(A), so that Axioms 3 and 3' are not satisfied.
Thus these axioms are independent in their respective systems.
To see that Axian 4 is independent, let the term 3-progression
denote an infinite arithmetic progression with common difference 3,
and define a function d on 2' by:
1 if A-I; .(A = [+if A 9 I and contains a 3-progression;. 0 otherwise.
Axioms 1 and 2 clearly hold, and it is routine to verify that Axiams 3,
3' and 5 are also satisfied. To see that Axiom 4 does not hold, let
- - - A = (1,4,7, . . .) and B 2 {2,5,8, . . .). Then d(A) = d(B) =
Finally, to show that Axiom 5 is independent, define a
function d on 2' by: if A contains two disjoint 3-progressions;
d(A) =
otherwise.
Again, Axioms 1 and 2 clearly hold, and it may be easily verified that
- --- Axioms 3, 3' and 4 also hold. I%% if we let A = {1,4, i, . . .J lJ {2,5,8 . . . and B = {1,4,7, . .' .} U {3,6,9, . . .}, then d(A) =d(B) =1, while d(AnB) =O. Thus d(A) +d(B) >l+d(~fl~), so Axiom 5 does not hold. \This concludes the proof. - The next two lemaas are extensions of Axioms 3' and 4,
following by induction on i: ,
Lemma 2.2. If d is a translation invariant density, then for all A and all i 2 1, d(A + i) = d(A).
Lemana 2.3. If d is a density, and {Ai) is a finite n I iyl e collection of disjoint sets, then Z d (Ai) i=l
To prove countable superadditivlty of a density, we first require the following monotone property. - f
Theorem 2,2, If d is a density and A L B, then- d_(A)_5
proof', By Axioms 1 and 4, d (A) 5 d-(A) + d (B \ A) I ~heodm2.3. If d is a density, and {Ai 3i-1- is a 0 m countable collection of disjoint sets, then Z d (Ai) 5 d( U ~i). i=l i= 1
Proof. Suppose on the contrary that there exists a countable a* ma m collection of disjoint sets {Ai >i=l such that Z d (Ai) > d ( U Ai) . n n i=l i=1 Note that since for all n, Zdbi) Sd( UA~)1, ,we have OD - - A- _ i=L 2d(Ai) 51. Sowemay let ;d(Ai) = d( UA~)+ 6, where i=l e i= 1 i-1 0 S -Then there exists an N such that d(A;) < 6 .$,.', 2 A > a a i=l 6 N d ( Ai-) +-T , ar;d by monotonicity , d ( U Ri) 2 d ( Ai) . This gives i= 1 i=l i= 1 ' N N Z d(~~)> d( U -Ai), contradicting finite superadditivity. Therefore, for
i-1 i=l aD a OD every collection (~~jof disjoint sets, Z d (Ai) 5 d ( U Ail . i= 1 i= 1 i=l
We now introduce the abstract upper density. Given a function I - d on 2 , define a function d on 2I by B(A) = 1 - d(z). - Definition 2.5. If d is a density, then the function d
is called the associated upper density. rn
Note that the upper density is defined in terms of the density
alone. The order relation of density to upper density is as follows.
Theorem 2.4. If d is a density, then for all A, d(~)5
Proof. Using Axioms 2 and 4, we have d (A) A d (A) + d 6)-
The monotone property also holds'for upper density. TfiiiXj 2.5. If a is an upper density,.and A G B, then -
- C- Proof. If A G B, then B G A, so by the monotonicity of - density, d (E) 5 d (A). Then d(~)= 1 - d (z) 5 1 - d 6)= d (B):-
The following is a "mixed density" -result+
Theorem 2.6.- If A fl B = 4, then d(A U B) (.d(A) + d(~). - proof.- Let At = A U B and B.' = B, so that At fl Bt = A.
By Axiom 5, d(A U'B) + d(E) = dh') + d4B') 5 la+ d(A' n Bt) = 1 + d(A).
Thus d(A1 + d(~)= d (A) + 1 - d 6)1 d (A U B) .
The next thGorem shows that adding (deleting) a set of upper
density zero to (fran) any given set does not alter the density of the .
given set.
Theorem 2.7. If d is a density and B is a set such that - dB) = 0, 'then for all *A, d(A U B) = d(A \ B) = d(~).
1 Proof. Let C = B \ A, so that C G B, A U B = A U C, and , A fl C = 4. Then by monotonicity and Theorem 2.6, &(A) i d(A U B) =
Furthermore, since A \ B G A and A Ft B 2 B, we-have d (A \ B) 5
We now show that the upper density of a finite set is zero. Theorem 2.8. If d is a density and F is a finite set, then d(~)= d(~)= 0.
Proof. If F is a finite set, then I. So if d is an - - asymptotic density, then by Axioms 2 and 3, d(FZ = 1 - d(F) = 1 - d(1) = - 0. We also have for a finite set F that for some k, I + k G F. So
- and the monotone property, d(F) = 1 - d(F1.9 1 - d(1 + k) = 1 - d(1) = - 0. With Axiom 1, this gives d (F) = 0. - In both cases, since d (F) L d (F) 1 0, we have d (I?) = 0.
Now we demonstrate the relation between asymptotic density and translation invariant density that was mentioned earlier.
Theorem 2.9. A translation invariant density is an asymptotic density . \ I
Proof. Let d be a translation invariant density. If A - B, then by Eemma 1.1 there exists an N such that # 3 A U (1, . . . ,N) = B U (1, . . . ,N). Then by Theorems 2.7 and 2:8, d(A) = d(A U (1, . . . ,N)) = d(B U (1, . . . ,N)) = d(B). Thus Axiom 3 holds, and so d is an asymptotic density.
The converse of the above theor&n is not- true-. An example of an asymptotic dens,'lty mat: is not translztforr-~antwl;33&+fen in Chapter 3. ,
In the following theorem we give properties of upper density corresponding to the axioms. Note that Properties and 5 are not .* - Theorem 2~10. If d is an upper density, then:
if 'A U B-=%II then-- d(A) + d(B) 11+ Z(A 0 B).
If d is an upper translation invariant density, then: - - - 3'. for all A, d(A + 1) = d(A).
Proof.
- 2. Since the density, of a finite. set is zero, d(1) =
- 3. First observe that A A B = A a. Then A - B &plies
- - - that A A 5 is finite, fr& which N 8. So using Axiom 3, we have
-k 2 - (d(x) + d(~)).BY Axiom 5, for all A, B, d(K) + d(g) 5 1 + d(A fl g). Then d(~)+ d(~) = 2 - (d6) + d6)) 2 2 - (1 +.d(zn =
--- Now A U B = I implies that An B = A U B = +, so by Axiom 4 we have d a& +a&. -~~F-Z(J+Oa> =l - U-L~Z 1 - - (d(2 + d(g)), and 1 + d(~fI B) 5 2 - (d(z) + d(B)) = d(A) + d(~). - 3 ' . First observe that- A + 1 = (A + 1) It {I ). For
xEA+l iff x#a+1 forany'YF or x=1 iff x-1K~; * or x = 1 iff x - 1 E A, or x = 1 iff x E (x+1) U {l}. Then by
Theorems 2.7 and 2.8, d (A + 1) = d (z+ 1). Therefore, using Axiom 3
- ' The next two lenunas are extensions of Properties 3' and 4,
and follow by induction on i. ' - Lemma 2.4. If d is an upper translation invariant density, 6 - - then for all A and all i 2 1, d(~+ i) = d(~).
n Lemma 2.5. If d is an up& derislty, and {Ai}i=l is any n a finite collection of sets, then 2 Z(Ai) 2 d[ Xi]. i=l - i=l
It follows from Theorem 2.8 (zero upper density of a finite
set) that the above lemma cannot be extended to countable collections 00 - - of sets. For Z d({i}&'= 0 < 1 = d(I) = . It is of i=1 interest to note that while for density we have countable superadditivity - -- - - so - >..- of collections of disjoint sets, for upper density we have finite but
not countable subadditivity of Sllections of arbitrary sets.
We can use finite subadditivity to show that the addition
(deletion)- of a set of zero upper density to (from) a given set does w not alter.the uppez density of the given set.
Theorem 2.11. If d is an upper density, and B is a set - - - such that d(B) '= 0, then for all A, d(~U B) = d(~\B) = d(~).
.4 - Proof. By monotonicity and PropertyPz, d(~)5 Z(A U B) 5
- - d (A \' B) = d (A) .
The next three theorems include stronger forms of Axiom 4, - Theorem 2.6 and Property 4. - Theorem 2.12. If d-'is a density and d(A fl'B) = 0, then
d(A) + d(B) 5 d(A U B). - Proof. Since d(A n B) = 0, we have d(A n B) = 0. Then f B using ~xim4 and Theorem 2.7, d(A U B) = d((A A B) U (A fl B)) I
OD, Corollary 2.1. If d .is a density, and {Ai) is a i=l --- - - countable collection of sets such that for i # j, d(A - inAj) 00 then 2 d (Ai) 5 d[ ; Ail . i=l i=l r\ proof. This follo& from the "above theorem like-the proofs of Lemma 2.3 and Theorem 2.3. - Theorem 2-13. If d is ,a density and d (A B) = 0, then
- proof. Let A' = A U B and B' = B. Then A'nB1= P 6 A \ B = A \ (A n B), SO by Theorem- 2.7, d(~'n B') 2 d(~).The proof of d(A U B) 5 d (A) + d(B) then follows like that of Theorem 2.6. - * For the second inequality, since A U (A U B) = I, we have -- -- by Property T and Theorem 2.11 that d (A) + d(A U B) 2 1 + d (A n , (A U B) ) = - 1 + :(B \ A) = 1 + d(B \ (A fl B) ) = 1 + Z(B). Therefore d(A U B) > -- ,I - d(A) + d(~)= d(A) + d(B).
Theorem 2.14. If d is a 'density and d (A U Bl = 1, then I
-- Proof. We have d (A n B) = 1 - d (A U B) = 0, so by
Theorem 2.12, d (A) + d (g) 5 d (A- U B) . The rest of the proof follows like that of Property 5.
------We can calculate or give inequalities for the abstract density
- - - -- of certain classes of sets, as the next two theorems show.
Theorem 2.15. Let A be such that for i # j , (A + i) fI
(A + j) -4, and let d be a translation invariant density. Then Examples of "sparse" sets meeting the-- requirements of the
above theorem are { f } and in2 3. -
Theorem 2.16. Let A = {kn) , for some k Z 1 , and let n= 1 - 1 d be a translation invariant density. Then d(A) 5 i;. and d(A) e t .
0 Proof. Let A. = A, and for i = 1, . . . ,k - 1, rlet
Q) 4- Ai = {kn + i} '= A. + i. Then by Akiom 3', d(Ai) = d(A) for n= 1 k-1 =i=1, . . . ,k- 1. Also, the Ai are disjoint and U Ai = I. Thus
k-1.- - k-1 i=0 f 4 by &ma 2.3, kd(A) = C d(Ai) Id( U Ai) = 1, from which d(A) 5 - . i=O i=O k b 1 Similarly, using Lemma 2.5, we get kEI (A) 2 1 and d(~)Z -k .
Definition 2.7. Let a density d be called a normal density
00 if for every arithmetic progression. A = {kn) with k 21, n= 1 - 1 d (A) = d (A) = .
An example -of a normal density is the usual :(lower) ,
------p-pp - - - asymptotic aensity. For an example of a non-normal density, consider
the set function p defined on 2' by: 1 if A-I;
p (A) =
-- - ltkybe easily verified that p is a translation invariant density.
Definition 2.8. The function $ is.called the discrete.
density.
It follows that the upper discrete density is given by
1 if A is infinite; - $(A) = i0 if A is finite.
w , Now given A = {kn) for some k 2 2, we have 1 - n=l P(A). = 0 < -k. < 1 = $(A), so that $ is a non-normal density. The example of-discrete density also shows that the conclusion ' of Theorem 2.15 need not hold for upper 8ensity. This can be seen by
letting A be any infinite set satisfying' the hypothesis of the theorem,
Furthermore, the example of fl shows that Theorem 2.7 is not 7 - 8 true if the hypothesis d (B) = 0 is replaced by d(B) = 0. For if A= {1,3,5, . . .) and .g=(2.4.6, . . .), then $(B) =.Of but
We conclude this discussion of the discrete die- &ty by giving its order relation to the ordinary asymptotic density. First the usual notation is defined.
-
Definition 2.6. If dl and d2 are iunctidns on 2'. ' then -. d, 5 d2 means that A, dl (A) I d2 (A) ; and dl = d2 means I for all that
Lemma 2.6. If dl and d2 are densities such that, dl 2 d2 , then - - - - Proof. For all A, d2 (A) = 1 - d2 (A) 5 1 - dl (A) = dl (A)
Theorem 2.17 If $ is the discrete density,
-Q , . * usual asymptotic density, then $ 5 6 5 6 5 $.
$(A) = 1, then AwI and 6(A) = 6(I) = 1." nus fl 5 6, and the
1 3 e 1. conclusion follows from Theorem 2.4 and the preceding lemma, To
We mew define the natural density and give some of its
,.:
Definition 2.8. ~irenadensity ,d, .letJ hfd = & - $9' {A I d(A) = d(A) 1. Then the natuqal dbnsitl - vd : ffd + [Of 11 is is defined by vd (A) = d (A) '2 ??(A) . we may write 9 for vd , when r "<' d is understood from the c%nte&
, '3 - - * - pp Theorem 2.18. If d is a density and A 4 % , then
-- - Proof, We have d(A) = 1 - d6) = 1 - d(A) = 1 - d(~)= - --- b d (A) , so A F Md . Also, v (z) = d(~)= 1 - d(A) = 1 - v(A).
Theorem 2.19. If d is a translation invariant density and
A f Nd , then A + 1 C ffd . wit+ v(A+ 1) = v(A). - Proof. we have d(~+1) = 1 - d(A +,I) = 1 - d(A+ 1) =
C The following is a mixed density and natural density result.
Theorem 2.20. Let d be a density, and let B€Nd. If
Y~(A0 B) = 0, then d(A U B) = d(A) + vd(B), and g(.AUB) =
Proof. Note #at vd (A fl B) = 0 is equivalent, to ' - d(A 3 B) = 0. Then by Axiom 4 and the first part of Theorem 2.12, d [A) + vd (B) = d (A) + d (B) 5 d (A U 8) 5 d (A) + T(B) = d (A) + vd (B) .
F'urthermore , using Property and the second part of Theorem - - 2.12, we have d (A) + vd (5) = d(A) + d (B) 5 d(~U B) 4 Z(A) + ~(BI=
We now show *at natural density is finitely additive. vd(~fl B) = 0, then A U B € ffd and vd(A U B) = vd(~)+ vd(B).
Proof. Using Theorems 2 -12 and 2 .4, and Property 4, we have
v &] + v (3) = d (A) + d (B) 5 d (A u B) 5 d(A U B) 5 Z(A)+ Z(B) = - v (A) + v(B1, which implies that d(A U B) = d(A U B) so that
Definition 2.6. Sets A and B afe called almost disjoint
(with respect to density d) if vd (A n B) = 0.
Note that by Theorem 2.8, disjoint sets are almost disjoint with respect' to any density.
Now it follows from Theorem 2.16 by the usual induction that natural density is finitely additive for collections of almost disjoint sets. Natural density is not countably additive, however, since if d
m m is any density, then Vd(.U {ill = 1 # 0 = Z vd({i)). There are some 1=1 i=l natural densities though, for which an approximation to countable additivity called the additivity property (AP) holds.
m Additivity Property. Let (Ai} be a collection of disjoint
i=l Ce sets in Nd . Then there exists a collection (Bi) with each m ca i= Q)1 Bi - Ai , such that U Bi E Nd and v(,U Bi) = z v (Bi). 1=1 ' i=l -- i=l ------
We may also formulate a weak additivity property (WAP) by replacing the requirement that each Bi -Ai by the condition that for - each i, d{Bi A Ai) = 0. Since the upper density of a finite set is zero, it follows that the additivity property implies the weak additivity property. It is not known if the converse is true. , In the next section, a class of natural densities is shown to have the additivity property. be a sequence, arid let M = (Q) be an Let = (wkj' infinite matrix. Thenk ultiplication of the sequence by the matrix results in a new sequence Mx = y = (yn), where yn = (hfx), =
ankXk M is said to define a r&l& in5Erixkthod of seability k' if the convergence of x to s implies the convergence of Mx to s.
The following well-known Silverman/Toeplitz conditions- are necessary and sufficient for a matrix (ank) to be regular.
i) Vk, ank + 0 as n + =;
ii) sup z /ankl < -; nk
One of the most familiar examples of a regular matrix method is the
where cnk = - - 0 if k>n. The usual (lower) asymptotic density is related to the Cesaro method as follows. Given a set A, let XA = (XA(k) ) be the characteristic sequence of A defined by L--pp> - x&) n Then Ah) = Z XA(k) , and -=Ah) 1 A - "x, (k) = k= 1 n n k=l n m
Z cnkXA(k) . So 6 (A) = lim inf = lim inf (cX~), . '
Furthennore, the discrete density is related to the infinite
identity matrix, denoted by J = (j*) . Given a set A, (JXA), =
Z jnkXA (k) = XA (n) , so f3(A) = lim inf XA (n) = lim inf (JX,) . .. k n = n New fe-an axbiw- i- matrix- M- +-&he
dM : 2' + R by dM(A) = lim inf (MXAIn A major aim of +is chapter . . - .-- n ' - is to characterize those matrices M for which % is a density.
Firstly, % is well defined if (MXAIn exists for all A
and all n; i.e., if Z ankXA(k) converges for all A and all n.
k OD This condition is equivalent to. C la*] < =, for all n. The k= 1 following definition is required for the proof.
Definition 3.1. Let a E R. Then a+, the positive part of - a, is defined by a+ = max{O,a), and a , the negative part of a, - is defined by a = rnaxi0, -a). - + - Note that a = a+ - a , la1 = a + a , and a+ = a iff
a 2 0. Also, if B an is a series with (bn) and (cn) the sequences
of its non-negative and negative terms respectively, then Z bn = Z an+ - and Z cn = - Z a, .
4 Theorem 3.1. In order for Z aAXA(k) to converge for all A f k - - - - and all n, it is necessary and sufficient that for all* n,
Proof. The condition is sufficient , since if Z 1 ihk 1 c -, k ,/- for all n, then for all n and all A, I Z ankXA(k) I 5 k
9 To see that it is necessary, suppose there exists an n such
that 2 lankl diverges. Then by a well-known theorem on absolute k + - + convergence, Z ank and =nk cannot both converge, If Z a* k k ,, k d&erges, define A k E A iff a& ,0;' so that- Z aJlb:Xa(k) = - k '2 ank+ diverges. If only Z ank diverges, define A by k f A iff k k - =nk -= Om. Then ankXA (k) =-Za* diverges. In either case, there k k exists an n and an A such that Z adA(k) diverges. 4 k The conclusion follows by the contrapositive.
The condition 1 Z = 1 is not constructed from the Cesaro - n dM is a density. A conjecture here- is that if dM is a density, then there exists a matrix MI = (%I) with lim Z ankl = 1, such that for all ' nk mother consideration in regard to the condition f' lim Z ank = 1 is given by the following theorem [4]. - nk --a. --- ' Theorem 3.2. Let d= (a*) be a matrix such that / + density. Then in order that %(A) = lim sup (MXAInr fof all A, . it is n necessary and sufficient.fhat lim L anR = 1- - - -- nk 4 proof. To see that it is necessary, first note that,-- - (MX~)~= Z afi - NOW if %(A) = lim sup(MXA), for all A, then k. - n lim sup Z a* = 1im sup (NX1ln = d(1) = 1. Since lin inf 2 ank = n =k n n k %(I)= 1 also, it follows that lim ,Z a* = 1. an k For the converse.-w let - 1 denote the unit sequence {l,l, 1, . -1, and observe thdt X- -- 1"-X : Then using the fact that A A mltiplicati_on by M is lineal;, and the hypothesis lim ,Z a* = 1, we - nk have %(A) = 1 - % (?i) = 1 lim inf (MX-) = 1 - lim inf (M (L - XA) ) = A n - n n 1 - lim in•’(ML - MXA) = 1 + lim sup (MXA - M&In = 1 + lim sup (wA),- n n n lim(M&), = lim sup (MXAIn . n. n In view of the above considerations, the convergence of the sequence of row sums to 1 is assumed in the following theorem. Some . 9 definitions aEe required first. such ~~ -- - M is essehtially non-negative means that lim Z a* = 0. nk +. r Theorem 3.3. If M = (ank) is a matrix with lim Z a* = 1 -rs / nk such that % is a density, then fi is regular and essentially 9 non-negLtive.- z * . . -- Proof. since 41 is well-aefined aensm7jif TneorePl 2.1 we have for all n that y 4-w. After the proof 'that M is k 't, b essentially non-negative, it wPl1 be shown that the absolute row sums' are uniformly bounded. To see that the column limits are zero, for any -- given k, let A = {k). Then (MXA) = Z aniXA(i) = %k , &d since ?a 1 J i .,'A C NdM with .v(A) = 0, it follows that lim a* = lim(~X~),= V(A) = 0. F a - n n To see that M is essentially non-negative, suppose on the - contrary that there exists 6 > 0 such that, Z a* 7 6 for infinitely k many n. Let nl be such an n, and choose finitely many k, say - . - el ' kl . (eo = 0) with M~:~< kg +1 (M~= 0). and an integer ni such that i-1 I ei - •˜. r, > 2 and anikj < 0 for j = ei-l + 1, . . . ,ei. m Also, there exists Mi > ke such that Z I anikl c ; and Ni 7 Mi k=Mi such that for all n 2 Ni - , 1 a+ 14.8 k= 1 - - - --+ NOW let 9 = fEltXZ, . . ,kelSel+lT . . . , e2t . Then for i > 1, Now we have < . . . < k 5 Mi - 1 and so by the Mi-1 + 1 4 kei-l+l i . Mi-1 @Y definition of A, Z anik X A (k) = C anikj ' Since k=Mi,l+l j=ei-l+l i - < 0 for j = ei-1 +1,... anik tit and Sikj ' ei 6 j~e~-~+l ' 6 , it follows that Z %ikj < -5 -. - Z j=ei-1+l - -- -. SO we have (mAIni5 fi - 6 + = - 6 , and &(A) = 8 2 8 4 lim inf[MXA), 5 - z<6 0, contradicting Axian 1. This proves that we - must have lim Z a* = 0; i.e., that M is essentially non-negative. nk - - r, a*+ as n+=. Also, Cank=Eank+ -a* =Z a*+-Z a* + k k k k k Z ad+ as - n + -. Thus lim Z 1 allkl = lim ank = 1 (by hypot%esis) , \ k nk nk so that sup 2 la*l c m. Since alp three Silverman/Toeplitz conditions nk 4 are now fulfilled, we conclude that M is @regular. I A partial converse to the above theorem will now be proved. A - A - - --- \ We begin with some definitions and lamas. Definition 3.3. M = (ank) is a non-negative matrix, written M 10, if for all n and all k, ank 2 0. - A non-nebative matrix M+ can be associated with an arbitrary matrix M = (ank) by defining M+ = (ank+). '1 ~efinition3.4. Two matrices M arid M' - are equivalent, written M E M' , if for all A, lim( (MXA), - (M'X~)~)= 0. n P Lemma 3.1. If M z M', then dM = %, . Proof. Using well-known properties of lim inf, we have for sequences xn and yn that lira inf(xn - yn) 5 lim inf xn -' n n lim inf yn 5 lim sup(xn - yn) . Now given A, dM(A) - %I (A) = n n lim inf(MXA), - lim inf(M'XA), , and since for all A, n n fim((MXA)n - (MtXAIn) = 0, it follows that dMCAF - dMldA) '0, and - - n so dM@) = 43, fA) . Lemma 3.2. If M is an essentially non-negative matrix, then MEM.+ Proof. Let M= (a*). For all A, (q),- (MXA), + = - - - Theorem 3.4 [41, If M is regular and essentially non-negative, ------then dM is an ,asymptotic density, but not necessarily a translation invariant density. Proof. From lenrna 3 .2 we have dM = %+ , so we may simply assume that M is a non-negative matrix. Then for all A, dM(A) = + 'p t lim inf (MXAIn = lin in•’ Z adA(k) 2 0, and dM(A) = n n k Z 5 Z = a* = 1, lim inf ank 'X A (k) lim id hnkXI(k) lim inf Z since M k - n3 n k n k 4 is regular. So Axioms 1 and 2 hold. Now suppose that A - B. Then there exists an N such that for all k 2 NI XA(k) = XB (k) . Let xn = Z (k) and yn *' Ir -- - Z ank . Since the column limits are zero, it follows that k=l N . . 0. lim Z ank = NOW it may be -assumed that %(A) E dM(B) r SO n k=l dM(A) - dM(B) = lim inf x, - lim inf yn 5 lim sup (xn - yn) = 0. That - - n n - n - is, %(A) = dM(B) , and Axiom 3 is proved. - - - -- For Axiom 4, let A n B = 4, and observe that in this case, ?AB = XA + XB . Then %(A U B) = lim inf (MXAm)n = n- - lim inf Z a* (XA (k) + XB (k) ) 2 lh inf Z ankXR (k) + lim inf Z ankXB (k) - n k n k n k - Axiom 5 is proved via Property 4 for upper density. It is - - sufficient to prove the latter, since if d (A) + Z(B) ? d(A U B) for - - - - all A, B, then for all A, B, 1 - d(~)+ 1 -d(B) = d(A) + d(~)L * Note that in genqal,- - XalAz = X X X X Observe Calso that in a-+ B - _AB--.. the proof of Theorem 3.2, it is only required that = lim inf(MXAjn n satisfies Axiom 2. Since this has already been shown, we may apply the result' - here. Then for all A, B, %(A U B) = lim sup (MXAl$ln = I1- lim sup z anl;(XA (k) + xB(k) - xA(k)xB(k) ) 5 lim sup I: ankXA (k) + n k n k lim aup Z skXB(k) + lim sup X - ankXA(k)XB(k). SincO M is non-negative n k n k - lim sup Z - ankxA(k)X,(k) r 0, so we have %(A U B) C n k - lim sup Z ankXA(k) + lim sup Z aax,(k) = %(A) + %(B), which was to n k n k - be shown. This proves that % is an asymptotic density. To see that an asymptotic density obtained from an essentially non-negative regular - matrix may not be translation invariant, consider the density % associated with M = (1 0 . . . + , . If A is thesetof 1 1 -0 o... 2 2 ------1 1 1 -0 0 o... r 3 3 3 ------ L - odd numbers, then for all n, (MXA), = 1 so that 43 (A) = 1. But for all n, (MXA+l ) = 0 so that %(A + 1) = 0. ------The above tl%qremis only a partial copverse to Theorem 3.3, since the question of necessary and sufficient conditions for translation invari'ance remains open. 4 I ,We now shaw that the additivityproperty holds* -- -A Definition 3.5 141, A density d is a eaur lar matrix method densxt~if there exists a regular matrix M such that % = d. We restate a conjecture mad? earlier as follows. If M is a matrix such that 41 is a density, then there exists a regular matrix Theorem 3.5. All regular matrix method demties have the additivity- property. i Proof. Let % be a given matrix method density. hen by Theorem 3.3, M = (Q) Aisessentially non-negative and regular. By Lemmas 3.1 and 3.2, we may assume' without loss of generality that. M is a non-negative regular matrix. Thus for all n, C la*/ = Z ank < 43. SO for all n we can k k find s(n) such that - < ~ 43 -- - 1- I a* < , and s(n + 1) > s(n). k=s (n) +1 OD Now given a collection (Ai} of disjoint sets in kdH , let i=l By natuk ai . the finite additivity of density, we have - i- for each j that U Ai C and that - i=l %' - @ lim sup Z a&UA1(k) +. . . +X (k)) n k Aj = lim SU~(M*X A~U... UAj " n Iq So fo; each j, there exists an ~(j)such that if n L N(j), *en t. and we arrange that For n > define p(n) to be such that N(p(n)) 5 n -- - N ip (n) + 1). Given any integer M, definition, for all n 2 no , p(nf 2 M + 1 > M. Thus lim p(n) n Now for each i, define Bi by Biz % hi\ 11, . . . ,s(tTfi + I))), so that for each fydpl (Bil = vd,(Ai) . Given n, if i > p (n) , then m Let B = U Bi . Then i=l (0 = lim sup C n k=l s (n) p (n) = lim sup Z a* Z XB. (k) n k=l i=l 1 5 sup C lh ank 2 XAi(k) n k=l i=l p (n) 1 5 lim sup I: ai + lim sup p0 , n i=l n p t n'l = limsup C, ai n i=l The last step follows since O 5 2 9 =- Z - d 4)-5 5 OD i=l i=l d ( 3 Ail 5 1 implies that Z ai is a converqent -series. _ _Nw-using_ i= 1 i=l ' m w countable superadditivity of density, we have Z v(Ai) = Z v(Bi) = - Q) a - OD i= 1 i=l 5 d (3i) 5 d ( U Bi) 5 d (B) 5 C v (Ai) . From this it follows that i= 1 i= 1 i=l 3 5 L L -- 0 m U Bi € bf and that US( U Bi) = Z v4(Bi). Thus the arbitrary - % ------. 1= 1 regular matrix method density has the additivity property. -v \ In the last, chapter, the .usual asymptotic (and translation invwiant) density was shown to be associated with the Cesaro 4 surmnability method. In this chapter, we study a density associated in a similar way with the Abel summability method. ~efinition4.1. A series 2 ak is -Abel summable to the value k b L if lim z akxK = L. This may be written z ak = L (~2. -1- k k Equivalently, a sequence (sk) is -Abel limitable to the value L if k lim (1 - x) Z skx = L. This may be written lim(sk) = L (A). x-tl- k . 4 Although Abel summability is not a matrix method, it can be related to a "semi-continuous matrix" of .the form , where x increases over some given interval (which may be changed by a transformation of variable). For a given sequence s = (sk), define Ms by his (x) = Z skfk (x) . Let sequence s = (sk), when it exists, is given by lim Z sk (xk -r xk+') = - x-4- k- - - - lim Ask), - x+1- -- Now let a function a be defined on 2I by =(A) = , lim inf kA(x) = lim inf Z XA(k)(xk - xk"). This function is well x+l- 1 k defined, since for all A and for 0 5 x < 1, we have emm ma 4.1. If a i.s as defined above, and is defined as - - usual by A = 1 - a (1, then a is given by =(A) = lim sup AxA (x) . - x+1- OD Theorem 4.1. The function a defined above is a translation invariant dendty. Proof. For all A and for 0 5 x.< I, we have o c c xA(k) (xk - xk+l) 5 (1 - x) z xk = (1 - X = x < 1. ~husfor k k all A, 0 i a(A) i 1, and Axiom 1 holds. OD Axiom 2 is satisfied since a(1) = lim inf Z (1 - x)xk = X x+l- k=l lim inf (1 - x)~= lim inf x = 1. x+1- x+1- For Axiom 3 ' , first note that XA+l (k) = XA(k - 1) (where XAfO) = Of. Then k = lim inf E XA(k - 1) (xk-1 - x ) x+l- k = a (A)'. - - - Then For xio om 4, let ~'nB = Q so that xAUB= XA + XB . = lim inf E xA(k) (xk - xk+l) 1 k k+l) lim inf z xA(k) (xk - x + lim inf Z XB (k) (xk - xk+l) 1 k x-tl' k = a (A) + a(B) . For Axiom 5, observe that for all A, XAuB . Then since lim sup E XAm(k) (xk - xk+') 5 lim sup 7, (xk - xk+') = 1, x+l- k x+l- k + lim inf E XB(k 1 k = a (A) + a (B) - lim sup z xAUB (k) (xk - &+l) x+l' k This concludes the proof. Definition 4.2. The function a of the above theorem is called -Abel density. .. J We have the follaring order relation. Theorem 4.2. If a. is the Abel density, and 6 is the usual - - density, then 6 5 a 5 a 5 6. I Proof. Given A, let o;, = Z XA (i) and let f (x) = 01 i=O z x,(i)xi (1 - X) xA(i)xi (XA(0) = 0). Then f (x) - i - i=O (1 - x) 1-x (Z X*(i)xi) (C xi) . A well-known theorem (81 states that if Z ai and i i E bi are absolutely convergent, and ci = a$i + albi-l + . . . + aibo , f then (x) the product series Z ci is convergent. Thus 2 = OD m (1 - x) i i z ( z xA(j))xi= z ioiix. i=O j=O i=O We have E(R) = lim sup on , so given E > 0, we can find ri n - - N such that for aff n l W, on < 6(A) + &-. Then - -7 00 < (6(~)+ E) lim sup (1 - x) I: ixi x+l- i=O OD X The last step follows since 2 ixl = 2 . Since the inequality i=O (1 - X) L - holds for all A and all e > 0, we have a 5 6. Then by the - - definition of upper density and Theorem 2.4, 6 5 a 5 a 5 6. It follows from the above theorem that hf6 & hf= . We actually have N6 = N, , as a consequence of the following well4cnown result in summability theory [61, which is given without proof. First note that if a sequence (s,) is Cesaro limitable to the value L, we write Theorem 4.3. If (sn) is a bounded sequence with L (A) f then limis,) = L (CF. The natural Abel density of a set A, if it exists, is the Abel limit of the sequence XA . Likewise the natural ordinary density of a set A, if it exists, is the Cesaro limit of XA . Since for all is bounded, Theorems 4.3 and 4.2 together give the following. A, XA Theorem 4.3. Natural Abel density and naturaL ordinary density are equivalent. That is, N, = N6 and va = us . ' - Since ordinary densiq has the additivity property SiKiz7 normal, it follows frwi the above - theorem that the same holds true for Abel density. $5 UNIFORM DENSITY We begin the study of uniform density- with the following generalization of the counting function Definition 5.1. If A is a set and m 5 n, then ~[m,nl denotes the number of integers in -A n {m,m + 1, . . . ,n3. - ,-Note that A[l,n] = Ah) , and that A[m,n] = (A fl [m,n]) (n) = A(n) - A(m - 1). We now introduce a sequence in terms of which uniform . I density is defined. Definition 5.2. For a set A and n 1 1, let an = an(A) = min A[m + l,m + n] rnl0 n Lemma 5.1. Given a set A, .let an be as defined above. Then 'lim an exists (and equals sup an). n n Proof. We first show that given N > 0 and E > 0, there exists an M such that for all n >- M, an > aN - s. To see this, let M > N be such that for all n 1 MI We have for all n *at for all m 0, so Also, by definition we have A[m + l,m + Nl 2 aN L N that for all n 3 N, Now for all n 2 M and for all m, Thus for all n 2 M, an = min Arb + 1.m +. nl - m n , and by the result already proved, there exists an '=no ' SupN ' - - ~>n~suchthat for all n?M, an>%O-%>sup%-~. SO N-. lim inf an 5 sup % - E. Thus we have sup % - E 5 lim inf an 5 n N N n lim sup a, I sup % . Since e was arbitrary, it follows that n N lim an = sup % . n N Theorem 5.1. Let an be defined as above. Then u defined on 2' by u(A) = lim an is a translation invariant density.. n % Proof. Po? all m 10, n 2 1, and all A, 0 I Arm + l,m + nl 5 I n, so for all n 1 1, 0 4 a, 5 1. Thus for all A, 0 5 U(A) 5 1, satisfying Axiom 1. Since for all m 1 0 and n l 1, I[m + l,m + nl = n, it follows that fok all n 2 1 q, = 1. Thus u (I) = 1, satisfying Axim For Axiom 3', observe that (A + 1) [m + l,m + n] = / ~[(m- 1) + 1, (m - 1) + n], and that A[O,n - I] = A[l,n - 11 = ~(n- 1). Since for all A, lim Ah- 1) - -A(n) =* ,o for all A, n n n lim rnin A[(m - 1) + 1,h - 1) + nl = lim min Arm + l,m + n] = v(A). n nP0 n n m?O n Therefore, v(A + 1) = lim min @ + 1)[m + 1,m + n] -= . . n I@EO n lim rnin A[(m - 1) + 1, (m - 1) + nl = u(A), n m2O n For Axiom 4, let A fl B -= I$ . Then = lim min Aht. l,m + nl -t B€m + 1,m + nl n m n 1 lim rnin Arm + 1,m + n] + rnin B[m + 1,m + n] n ( m n m n ) Now given any A and B, we have U(A) + V(B) = lim min Arm + 1.m + nl + Jim min B[m + l,m + n] n m n n m n = lim min,(A U Blfm + l,m + nl + (A fl B)[m + l,m + n] n m n (A il ~)[m+ I,m + nl 5 lim mint1 + n 1 n m = 1 + lim min (A n B) [m + 1,m + nl n m n - Definition 5.3. Let an be defined as above. Then the function u defikd on 2I by u (A) = lim an(A) is called uniform n density . We now give an explicit form of upper uniform density. - - Theorem 5.2. For a set A, let an = %(A) = max Arm + l,m + nl, If u is the uniform density, then is given by ma n- E(A) = lim an . Proof. For all A, - Since the limit exists, it follows that lim exists, and v (A) = A sequence (sk) is said to be almost convergent to L if m+n m+n 1 lim Z sk = L, uniformly in m. Note thbt Z XA(k) = n k-1 k=m+l Afm + 1,m + n]. Thus a set A has natural uniform density a if its characteristic sequence XA is almost convergent to a. Uniform density is our first example of a density without the additivity property. Before proving this we give the following .- definition and lemma. Definition 5.4. ATblock in set A is a set - fk,k + 1, . . . ,k + n) 2 A, and a =in A is a block in A. The length of a block (gap)' is the number of integers in the block (gap) . LePPma 5.2. If set A has arbitrarixy long gaps, then - - v (A) = 0, Tf -A has arbitrarily long blocks, then V (A) = 1. Proof. If A has arbitrarily long gaps, then for all n, ------*% % = 0, SO ufAJ = 0, If A has arbitrarily long blocks, then A has - C arbitrarily long gaps, so v (A) = I - v LA) = 1. Theorem 5.3. Uniform density does not have the gdditivity property . OD Proof. Define a collection of sets {Ai) by {All' {2nL i-1 i= 1 and for i > 1, A = A + i-1 U Aj . The? the Ai are j=1 disjoint. If m + 1 5 2k 5; m + n, then log2 (m + 1) 5 k 4 log2(m + n) , [m ++ 1,m + nl [log2 -1 + -c [1cg2 n]. Thus for all n, max n = - m- O. -5A1(n) [log2nl so u(%) = ~henu~ingthemanotonepraperty, n n I and tt.le fact that u is a translation invariant density, we have for all i that ;(A~) = 0. Thus each of the Ai has natural uniform density zero. m Now let (B~) be any collection of sets such that Bi Ai i=l - for each i. Since u is an asymptotic density, u(Bi) = 0 for each i. - m Given any m, U Bi has blocks of length m. To see this, i= 1 observe that by Lemma 2.1, for i = 1, . . . m there exists N(i) such that Bi\ {I, . . . i}= A 1 . . . N.So if N = i=l,... ,m A 1, . . . N}. Thus for n such that 2" r N, we ,have m Now since U Bi has arbitrarily long blocks, by Lemma 5.2, additivity property. We now give the order relation of uniform density to ordinary - density. Theorem 5.4. is the uniform density, and is the - - ordinary density, then v 5 6 5 6 5 v, and Nu c kb with v6 - vV on NV Ermf. Ford- ~t,-mimRfiPrT.IiiiL+ nl --;~m;yfor all -0 n - - n A, U(A) C 6(~).Then by Lemma 2.6, v 5 6 5 6 5 v. With these inequalities and the definition of natural density, it is clear-at if then and vg (A) = . vv (A) . Now let A be defined by XA = {1,0,0,1,1,0,0,0,0,1,1,1,0, . . .), where at the nth step, n consecutive ones are followed by 2" e A (n) consecutive zeros. Maximum values of occur when n is the final j -n integer of a block of A. The formula for these integers is - "(1 + n) + 2" 2. Then 2 - 6(A) = lhn sup -A(n) = n n lim sltp ~[5(1+ n) t 2" 21 = lim sup - f(i+ 1;) =o, so n n 2(1J+ n) + 2n - 2 n -(1n2 + n) + 2n - 2 But since has arbitrarily long gaps and arbitrarily long - blocks , by Lemma 5.2, v (A) = 0 and v (A) = 1, SO A 6 NU . T~US The existence of natural uniform- -- density_-- cxuld-bekaken- aca ------ definition of a set being "uniformly distributed. I' Then we woulhexpect- --- arithmetic progressions to have natural uniform density, as the next result shows is the case. Theorem 5.5. Unifonn density is a normal density. Proof. Let A = {kn) for some k L 1. For all m, n=l ( (n + 1) - 1 - 5 ~[m+ l,m + nl 5 SO vU(A) = lim an = lim an = k -k* n n t To conclude this chapter, we calculate the natural uniform t density of another class of sets. ------Definition 5-5. Let a set A be an increasing sequence a.Then A is a lacunary -set if lim - an= -. n r Theorem 5.6. The natural uniform density of a lacunary set is zero. Proof. Let A = (an) .be any lacunary set. It is sufficient d 0 - 1 1 to prove that u (A) = 0. Given E 7 0, let K be such that - < '6, K i1 and let N be such that for all n L N, an+l - an > K. Let ni = a~ + ik - % + iK. For all m, Arm + l,m + nil 5 N + so ani (A) = K Afm + 1,m + nil - max f N + -1 . men ;(A) = lim an(A) = m "i %+iK K n - 1 - lim ani (A) 5 < E . Since E was arbitrary, v (A) = 0. i LOGARITHMIC DENSITY \ Our final density is a regular matr'4 method density. It is derived from the logarithmic summability meqod: a special case of a Riesz typical mean [6]. Let L = (ank) , where - "*- if k > n. Note that We now give a sufficient condition on a matrix M in order for d to be a translation invariant density. Theorem 6.1. If M = (%) is an essentially non-negative \ with = 0, then dM is a regular matrix lim 2: 1 a - an, k+l I nk n,k- translation invariant density. Proof. By Theorem 3.4, % is an asymptotic density. To see ------that dM is translation invariant, firstnote that XA+l(k) =- XA(k - 1). Then Corollary 6.2. The function X = dL defined above is a translation invariant density. Proof. The matrix L = (ank) defined above is non-negative. It is regular, since for all k, lim a& = 0, and lim Z a* = n n nk lb lim -1 Z 1 = 1. Finally, Z lantk - %,k+ll = n ~nnk=l k nk lim 1 n-1 1 1 =lim 1 =O. So by the above theorem, -(k=l - - -+ -) -. n Inn k k+l n n Inn X = dL is a translation invariant density. Definition 6.1. The function X defined above is called the logarithmic density. - - - - - -- We note that by Theorem 3.2, the upper logarithmic density - - X is given by (A) = lim Sup -1 Z 1 ; and by Theorm 3.5, n Inn afA a a5n logarithmic density has the additivity property. Since the usual density normal. meorem 6.2. If 6 is the usual density, and. X is the logarithmic density, then 6 5 X 5 5 K. n Proof. Given sequences (xi) and (yi), with sn = Z xi , n+k n+k i=1 me AbeI surmna€iori fSmiUia is Z xiyi = Z Ti @i - yi+l) - i=n+l i=n+l 1 Snyn+l + sn+kyn+k+l Given A, let xi = XA(i) and yi = 0 . -i s 7 that sn = A (n) . Then given M > 0, Since ---A(n) A(N) is bounded, we have n N+l 1 XAW XA(i) X(A) = lim inf -( Z +C- 1 n Inn i=l i i=N+1 i n- 1 1 XA(i) A(i) = lh inf =( .-. 1 + - 1 i+l+0(1)) 4 - n i=l i-1 - -- - . - n-1 = inf -1 C A (i) 1 Inn -.-i n i=N+1 i+l ,- inf -A(i) lim inf -1 G ,. 1 iH i n Inn i=N+1 i + 1 = inf -A(i) hN i -. Then letting N + =, we get X@) 2 lim inf -fi--A(n) = 6(A). It "- - follows by Theorem 2.5 and Lemma 2.6 that 6 5 X 5.X 5 6. By the above theorem, Ni L Nh . We actually have N6 C NX , since sets of the following type have natural logarithmic density but may not have natural ordinary density. Definition 6.2. Let A = (a,). A is a primitive -set if for We state the following results from Sequences [51 without - proof: If A is a primitive set, then 6(A) = X(A) = X(A) = 0, .and 0 5 6(A) < - . Moreover, given any E > 0, there is a primitive set ?4 1 A such that F(A) > + - E . Together with Theorem 6.2, this gives the following. Theorem 6.3. If 6 is the usual- density, and- h is the logarithic density, then Nh c Nb , with vX = V6 on Ng . $7 CONCLUSION * %' !r In this chapter we mention some other generalizations of ' 8 t asymptotic density, and give some applicagions of the density concept in sumability theory and number theory. We restate the unsolved problems that have been mentioned, and give some new ones as well. . ------In this paper, a general asymptotic density has been defined axiomatically, using as axioms some of the elementary properties of ordinary asymptotic density. Asymptotic density for sets of integers, and for arbitrary sets, has been generalized along more measure-theoretic lines by Buck [1,21. If sets- A and B have natural density according to his definitions, then so do A U B and A fl B; this does not hold for natural ordinary density [ll . ~sym~toticdensity has also been, generalized to sets of n-twples of integers by Freedman [31. The density concept has been applied-in summability theory by Freedman and Sember [41. Given an asymptotic density d, a sequence 1 b x is said to be (d)-nearly convergent to the real number L, if there \ exists an A with d(~),= 0 such that, when the terms of x indexed by elements of A are deleted, the resulting sequence converges to L in the ordinary sense. t The set Wd of (dl-nearly converqent sequences is a linear - - -- space, and can be used to describe the convergence fields and strong - convergence fields of summability methods related to d. Whether or not d has the additivity property for sets of natural density zero affects the results here. Some applications of the general theorems are that if 1 ol 1 is the sp&e of strongly Cesaro-sumable seipences, 1 AC 1 is the space of' strongly almost-convergent sequences, and m is the space of bounded sequences, then lol I fl m = W6 fl m, and ~AC = 1 i i- (ACI fl m 3W, fl m, with IAC~ =W, fl m . ' One of the usesl.of asymptotic density in number theory is in the concept of "almost all," similgr to the use of "aImost-alLYh--the - + - sense of measure. A property is said to hold for almost all integers if the set of integers for which the property does not hold is of 8 natural ordinary density zero. An example of a result stated in this t form is that almost all integers are the sum of a kth power and a prime [71 . Clearly, the definition of "almost all" may be modified to "(d)-almost all" for a general density d. If /dd0 denotes Me class 0 0 of sets A with vd (A) = 0, then ff, c N6 , so for example, the i statement that P holds for (0) -almost all integers is stronger, than the statement that P holds for (6)-almost all integers. An example of a general theorem in additive number theory - involving ordinary asymptotic density is Ostrnann's result that if 6(A) + 6(B) > 1, then A + B -. I 171. There is also an important theorem bp Kneser, which states that under certain conditions, 6(A + Bf 2 6(A) + 6(B). [51. It may be asked if these results hold fqr \ the generalized density d. - b i C - other unsolvd prahlems in'the theory of itsyntptetic denskty - 7 \ \ presented here are; i -- - method? That is, given a density d, is there a method M such that lim XA = a U.Il if and only if A C Nd with vd(~)= a. (iil Is every matrix-method density a regular matrixaethod density? -- - (iiT) Is fie we& additivity property e@ivaient to Ule additivity property, or do there exist densities which have the WAP but not the AP? In particular, does uniform density have the WAP? (iv) Find a necessary and sufficient condition on a matrix M = (ank) in order for dM to be a t'ranslation invariant density. We note again that a sufficient condition is (vIt Is it true that for a32 A and B, - d (A) + d (B) f :(A U B) + d (A fl B) ? (vi) Find a set A for which 6 (A) < a(A) . Since - - 6 5 ar 5 a 5 6, it is necessary that A XN6 . - (vii) Is a 5 X? We cannot have X I a 5 a IXI since then ,- the existence of natural logarithmic density would imply the existence of natural ordinary density. The known ordering of asymptotic densities %- that have been studied here is / hrrthennore, vu , v6 , and v~ are proper extensions of vf~I vu I and v6 respectively. i Buck, R. C,, The measure theoretic approach to density, Amer. J. Math., 68 (1946) , 560-580. Buck, R. C., Generalized asymptotid density, Amer. J. Math., 75 (l953), 335-346. Freedman, A. R., On asymptotic density in n-dimensions, Pac, J. Math., 29 (l969), 95-113. * - -- - A -- -- - Freedman, A. R. and Saber, J. J., Abstract densities and strong summability, unpublished manuscript. Halberstam, H. and Roth, K. F., Sequences, Vol. 1, Oxford univ. -, Press, 1966. Hardy, G. H., Divergent Series, Oxford Univ. Press, 1949. Niven, I., The Asymptotic Density of Sequences, Bull. Amer. Math. SOC., 57 (1951), 420-434. Taylor, A. E. and Mann, W. R., Advanced Calculus, Xerox College Publishing, 1972.