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Unformatted text preview: r of basic mathematical contexts
including the Bolzano Weierstraas theorem and walks in
Let f:Z+ Æ Z+ be strictly increasing. We define f#:Z+ Æ Z+ by
f#(n) = ff...f(1), where there are n f’s.
We define the Ackerman hierarchy as follows. Take f1:Z+ Æ Z+
to be doubling. Take fk+1:Z+ Æ Z+ to be (fk)#.
Note that f2 is base 2 exponentiation, and f3 is base 2
BW THEOREM. Let x,x,... be an infinite sequence from
the closed unit interval [0,1]. There exists k1 < k2 < ...
such that the subsequence x[k1],x[k2],... converges.
BW WITH ESTIMATE. Let x, x,... be an infinite sequence
from the closed unit interval [0,1]. There exists k1 < k2 <
... such that |x[ki+1]-x[ki]| < 1/ki-12, i ≥ 2.
THEOREM 11. Let r >> n ≥ 1 and x,...,x[r] Œ [0,1]. There
exists k1 < ... < kn such that |x[ki+1]-x[ki]| < 1/ki-12, 2 £ i
In the r >> n above, how large must r be relative to n? If n
= 11 then r > f3(64) = an exponential stack of 64 2’s. fn8(64) < r(n) < fn+c(n+c), for some universal c, n ≥ 10. In
fact, it is outrageous earlier than n = 11. We are looking to
see just when.
Let k ≥ 1. A walk in Nk is a finite or infinite sequence
x1,x2,... Œ Nk such that the Euclidean distance between
successive terms is exactly 1.
A self avoiding walk in Nk is a walk in Nk in which no term
Let x,y Œ Nk. We say that x points outward to y iff for all 1
≤ i ≤ k, xi £ yi.
THEOREM 12. Let x Œ Nk. In every sufficiently long finite
self avoiding walk in Nk starting at x, some term points
outward to a later term which is at least twice the Euclidean
distance from the origin.
Now let W(x) be the least n such that:
*in every self avoiding walk in Nk of length n starting at x,
some term points outward to a later term which is at least
twice the Euclidean distance from the origin*
THEOREM 13. W(2,2,2) ≥ 2192,938,011. W(1,1,1,1) ≥
E*(102,938,011). $c,d > 0 such that "k,n ≥ 1, A(k,n+c) £
W(n,...,n) £ A(k,n+d), where there are k n’s.
6. BLOCK SUBSEQUENCES.
For each k ≥ 1, let n(k) be the length of the longest finite
sequence x1,...,xn such that no consecutive block
is a subsequence of any other consecut...
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