Set Densities From Finite L.d.a. Element Sets

We'll present a series of tables and a figure to explain in greater detail how the integer densities in the collection of subsets obtained by detailed dissection of the l.d.a.s in the abstract tree lead to the conclusion that all odd positive integers appear in it.

First,  we had better be sure we understand how the elements of the l.d.a.s which share a d(=2^a*3^b) value map into the cardinality table which provides the basis for summation of the appropriate densities to determine the total density of odd integers in the tree.  The following table expands the upper left 3 by 3 corner of the cardinality table by indicating the l.d.a.  elements contributing to the cardinality.  The l.d.a.s appear in e(s|b)kt notation and the depth of each element in its l.d.a.  appears in parentheses.

Note that the et , est,  ebt,  and esst l.d.a.s are the only ones whose entire contents are presented,  because the later elements of the other l.d.a.s mentioned contain a>6,   outside of the pictured fragment.  The bracketed d values define the density of integers as 1/d  (among all integers)  or 2/d  (among only the odd integers).  As is basic to the very existence of the cardinality table,  the number of entries in each box determines the entry in that position in the cardinality table.  Thus,  the entries in each box constitute the complete set of odd integers contained by the set of l.d.a. elements sharing a d value in the finite interval denoted in brackets.

The density contributed within the intervals in this first table establish the densities these l.d.a.  elements will provide within larger intervals provided that the larger intervals are integer multiples of the initial ones.  Fortunately,  the widths occupied by the sets of sets as a is increased across the rows of the cardinality table do always arise as multiples by 2 in the row dimension or,  when b is increased in the column dimension,  as multiples by 3.  We will synchronize the intervals covered by each subset by starting them all at 3,  thus assuring that all of them appearing in any given row or column will come to a common terminus at the end of the largest of them.  To underscore this point,  the densities are expressed in unreduced fractions in Tables 2, 3, and 4.   The use of unreduced fractions reflects the patterns initiated by,  and evident in, this first table and demonstrates the set width growth by multiples of  2 or 3 as required to justify the summation of densities in finite sets.


Table 1. L.d.a. Element Contents of Cells for 3<=a<=5 and 1<=b<=4

from a=3 from a=4 from a=5 ---------|------------------|---------------------|------------------------------------------ from b=1 |et(1) 21[24] |est(2) 3[48] |ebt(2) 81[96] esst(3) 87[96] ---------|------------------|---------------------|------------------------------------------ from b=2 |est(1) 5[72] |esbt(2) 115[144] |essst(3) 23[288] essbt(3) 151[288] |ebt(1) 61[72] |esst(2) 131[144] |ebst(2) 113[288] ebbt(2) 241[288] ---------|------------------|---------------------|------------------------------------------ from b=3 |ebst(1) 85[216] |esbbt(2) 19[432] |esssbt(3) 119[864] ebsbt(2) 497[864] |esbt(1) 173[216] |essst(2) 35[432] |essbst(3) 247[864] ebbst(2) 625[864] |ebbt(1) 181[216] |essbt(2) 227[432] |ebsst(2) 305[864] ebbbt(2) 721[864] |esst(1) 197[216] |esbst(2) 355[432] |essbbt(3) 343[864] esssst(3) 791[864] ---------|------------------|---------------------|------------------------------------------ from b=4 |esbbt(1) 29[648] |esbsst(2) 67[1296] |ebssbt(2) 17[2592] essbbst(3) 631[2592] |essst(1) 53[648] |esssbt(2) 179[1296] |ebbsbt(2) 49[2592] esssbst(3) 695[2592] |ebsst(1) 229[648] |esbsbt(2) 211[1296] |essbbbt(3) 55[2592] ebbsst(2) 1201[2592] |essbt(1) 341[648] |esbbst(2) 307[1296] |ebbbst(2) 145[2592] ebsbbt(2) 1649[2592] |ebsbt(1) 373[648] |essbst(2) 371[1296] |ebsbst(2) 209[2592] essbsst(3) 1687[2592] |ebbst(1) 469[648] |essbbt(2) 515[1296] |essssst(3) 215[2592] esssbbt(3) 2135[2592] |esbst(1) 533[648] |esbbbt(2) 1027[1296] |essssbt(3) 503[2592] ebbbbt(2) 2161[2592] |ebbbt(1) 541[648] |esssst(2) 1187[1296] |essbsbt(3) 535[2592] ebssst(2) 2321[2592]

We have elsewhere presented tables illustrating the details of the downward and upward stages in the dissection of the elements in each branch of the abstract tree into disjoint subsets which each represent the contents of some one single element of some one l.d.a.  But those tables do not make it immediately obvious how the densities of the dissected subset are reduced in both the downward and the subsequent upward processes.  Accordingly, an additional diagram is now presented in which the width of the subsets represent their densities. Although the scaling of the top row differs from that of the other two rows and the depth reached in its development of the abstract tree is very limited,  the diagram makes it clear that the subsetting occurs during both directions of travel,  resulting in the very fine dissection which is shown in the above table.

It remains to show that the densities represented by all the sets of elements of all the l.d.a.s indicate that all the odd integers are included in the abstract tree.  To this end, three additional tables will be presented.  Note that the densities employed are 1/d (appropriate to consideration of odd positive integers only) throughout.  We will be able to present data describing a somewhat more liberal selection from the space of the cardinality table -- a 7 by 6  (or 5 by 5)  corner from the upper left.

Table 2 shows the density among the odd integers of integers provided by every  (a,b)  pair.  Remember that the densities cited are those defined within the finite  (2^a*3^b)  width of the first instance of the contained sets of  l.d.a.s elements,  but that,  conveniently enough,  later sets of elements do occur with a width obtained by multiplication by 2 or 3.  The densities are presented as fractions which have not been reduced to lowest terms,  because their pattern is more easily discerned in that way.  These densities are calculated from the formula involving the Fibonacci series (F(i)).
Using 2b-a*F(a-2)/3b, the following table results.

Table 2. Density of Odd Integers for Cells 3<=a<=9 and 1<=b<=6

b\a 3 4 5 6 7 8 9 1 1/12 1/24 2/48 3/96 5/192 8/384 13/768 2 1/18 1/36 2/72 3/144 5/288 8/576 13/1152 3 1/27 1/54 2/108 3/216 5/432 8/864 13/1728 4 2/81 2/162 4/324 6/648 10/1296 16/2592 26/5184 5 4/243 4/486 8/972 12/1944 20/3888 32/7776 52/15552 6 8/729 8/1458 16/2916 24/5832 40/11664 64/23328 104/46584

Table 3 shows the row-cumulative values of the densities across a constant 3^b value.  Again,  fractions which have not been reduced to their lowest terms are employed.  The cumulative densities from a to at in row b are given,  again employing the Fiboniacci series,  by using

add(2b-a*F(a-2)/3b),a=a..at    #row cumulative
the following table results.

Table 3. Cumulative Row Sums of Density for Cells 3<=a<=9 and 1<=b<=6

b\a 3 4 5 6 7 8 9 asymptote 1 1/12 1/8 1/6 19/96 43/192 47/192 67/256 ... 1/3 2 1/18 1/12 1/9 19/144 43/288 47/288 67/384 ... 2/9 3 1/27 1/18 2/27 19/216 43/432 47/432 67/576 ... 4/27 4 2/81 3/81 4/81 19/324 43/648 47/648 67/864 ... 8/81 5 4/243 8/243 8/243 19/486 43/972 47/972 67/1296 ...16/243 6 8/729 12/729 16/729 19/729 43/1458 47/1458 67/1944 ...32/729

The values calculated by formula for at=100 and at=200 had approached the asymptotic value to about 56 parts in 10^11 and 23 parts in 10^20, respectively.  Finally,  from these asymptotic values, the sum((2^(i-1)/3^i),i=1..infinity)  is 1,  showing that all the odd integers are included in the asymptote based on increasing widths of finite sets.

Finally, the Table 4 (showing a 5 by 5 chunk of the territory)  shows a column-cumulative value for each cell from the cardinality table.  Using

add((2b-a*F(a-2)/3b),b=b..bt)  #column cumulative
the following table results.

Table 4. Cumulative Columns Sums of Density for Cells 3<=a<=7 and 1<=b<=5

b\a 3 4 5 6 7 1 1/12 1/24 2/48 3/92 5/192 2 5/36 5/72 10/144 15/288 25/288 3 19/108 19/216 38/532 57/864 95/1728 4 65/324 65/648 130/1296 195/2592 325/5984 5 211/972 211/1944 422/3848 633/7776 1055/15552 asymptote: 1/4 1/8 2/16 3/32 5/64

The values calculated by formula for bt=100 and bt=200 had approached the asymptotic value to about 13 parts in 10^19 and 16 parts in 10^37,  respectively.  These asymptotic values constitute the expression sum((F(i)/2^(i+1),i=1..infinity)  whose value is known to be 1,  in agreement with the result from the row sums,  and,  once again,  indicating that in the asymptote all odd positive integers appear in the abstract tree.

If you prefer MapleV4's opinion to my "engineering" approach, consider the following fragment from a MapleV4 session.  The same expressions used above to gather the row-cumulative and the column-cumulative finite sums of the densities in the cardinality table are used again, but this time to determine the limit to the contribution of the densities as the initial row/column is increased to infinity.  Once again,  it appears that in the limit all odd positive integers appear in the abstract predecessor tree.

> f := proc(n) option remember;
>  if n<2 then n else f(n-1)+f(n-2) fi end;

    f := proc(n) option remember;if n < 2 then n else f(n - 1) + f(n - 2) fi end

> a:=3; b:= 1;

                                a := 3
                                b := 1

> limit(2^(bt-a)*f(a-2)/3^bt,bt=infinity,left);  #row cumulative

                                  0

> limit(2^(b-at)*f(a-2)/3^b,at=infinity,left);   #column cumulative

                                  0

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