Copy Theorem

This interactive demo explains the 
period copying phenomenon.
In order to understand this demo, you should
first read about the  the Box Theorem .

 Applet Instructions 
The   applet  at right is an enhanced
version of the one we used for
our explanation of the Box Theorem.
Activate the applet by pushing the
red arrow keys.  In general,
click on the arrow keys to trigger
a new computation.

You can select the orange fraction
using the mouse buttons and the
keyboard in tandem.  The fraction
you select must be odd/odd, and
have denominator at most 99.
The red fraction is controlled
by the arrow keys, in a manner
explained below.

The picture in the bottom window
is scaled with left and right
mouse clicks.  If you don't have
a 3-button mouse, or if the applet
doesn't seem to work properly with
your mouse, you can use the 3 button
mouse emulator on the applet.

 Diophantine constants 
Say that a rational p/q is  odd  if pq is odd.
Suppose A1=p1/q1 and A2=p2/q2 are two odd rationals
with p1 less than p2.  The  Diophantine constant  
D=D(A1,A2) is defined as the largest integer K
such that 

|A1-A2|<2/(Dq_1^2).

This notion makes
sense for any pair of rationals, but the applet
deals exclusively with pairs (A1,A2), both odd
rationals, such that |p1q2-p2q1|=2.  In this case,
we call the pair (A1,A2) an  odd Farey pair .
The terminology comes from the close connection
between our constructions and the Farey graph in
the hyperbolic plane.

The first thing the    applet  does is
allow you to sample some Farey pairs.  Once you
specify A1 (the orange fraction) you click the
arrow keys to generate A2 (the red fraction).
The value next to the arrow keys is the
Diophantine constant of the pair.  Given A1,
there is a unique choice of A2 for each positive
integer k.  The applet lets you sample up to
k=12, for all choices of A1, where both p1 and q1
are less than 100.  When you click the arrow
key, you trigger a computation of the arithmetic
graphs corresponding to A1 and A2.  We will explain
the significance of these graphs in the next 
section.

 Period Copying 

let G1=G(p1/q1) and G2=G(p2/q2). In general,
we will let X1 stand for the object of the
form X(p1/q1).  Likewise for X2.

In the discussion of the Box Phenomenon, we
explained that one period of G1 is
contained in the union of the two 
parallelograms F1 and S1.  Other
periods of G1 are contained in
translates of F1 and S1.
Thus, we can cover arbitrarily large
stretches of G1, either going to
the left or to the right, by stringing
out these parallelograms in succession.
If you click on the arrow keys of the
applet, you will see what we mean.

Let LEFT1(n) denote the union of n
of these parallelograms, starting with the
ones whose bottom right corner is the 
origin, and moving to the left.  Likewise
define RIGHT1(n).  We prove the
following result in the monograph.

Period Copying Theorem   Suppose
that (A1,A2) is an odd  Farey pair
with Diophantine constant k at least 1.
If A1 less than A2 then A2 
copies the portion of A1
that is contained in RIGHT1(k).  If
A1 greater than A2 then A2 
copies the portion of A1
that is contained in LEFT1(k). 

 The Halfbox Version 
If (A1,A2) is an odd Farey pair with
Diophantine constant 1, then we have
an alternate result.  Recall
that R1 contains a single period 
of G1, starting from the origin and
moving right.  Let R1' denote the
translate of R1 that contains the
period of G1 that starts from the
origin and goes to the left.  Let H1
denote the result of scaling the union
of R1 and R1' about the origin by a factor
of 1/2. So, H1 has the same width
as R1, but is half as tall. We
call H1 the halfbox.  you can draw
this set on the applet by toggling
the halfbox button on the display panel. 

 Halfbox Theorem   Suppose
that (A1,A2) is an od Farey pair
with Diophantine constant 1.
Let G1' denote the connected
component of G1 that contains
and origin and remains inside H1
Then G2 copies G1' as long as p1 
is sufficiently large 

Playing with the applet, one can see
that the result holds true as long as
p1 is greater than 1.  In the monograph
we only prove the result for p1 large.
This suffices for our purposes.

We would like to say more simply that
G2 copies the portion of G1
that is contained in H1, but we
do not prove this stronger statement.
What we prove is that 
G2 copies G1 until it leaves
H1.  The stronger and simpler
statement is probably true.

 Needed results 
The results above play different
roles in the proof of the main theorem.  

The Period Copying Theorem is first
used to prove that the Father-Son
decomposision result holds true.
We give an inductive argument.

Once we know that the Father-Son
decomposition result holds true,
we get the following corollary

 Period Copying Corollary   Suppose
that (A1,A2) is an odd Farey pair
with Diophantine constant at least 2.
If A1 is less than A2, then G2 copies at
least one period of G1, starting at the
origin and going to the right.
If A1 is greater than A2, then G2 copies at 
least one period of G1, starting at the
origin and going to the left. 

The Halfbox Theorem and the Period Copying
Corollary are the two results that feed into
our main argument, a geometric limiting
argument, that establishes the main theorem.

 Alternate Versions 
In the monograph we also prove versions
of the period copying theorem for
more general pairs (A1,A2) of odd
rationals.  For instance, 

 Weak Copy Theorem   Suppose
that (A1,A2) has Diophantine constant
at least 2k+2.  Then G2 copies at least
k consecutive periods of G1, starting
at the origin. 

 The case of even rationals 
The fact that A2 is an odd rational is not
important in the results above. The same
results hold if A2 is an even rational.
However, if A1 is an even rational, the
results are quite a bit different. They
must be different, because G1 is
a closed polygonal loop and G2
is always embedded.  It is impossible
for G2 to wind many times around
this loop and remain embedded. Something
different happens, and we leave it
to the reader to explore it.  We
ignore the cases involving even rationals
because these cases don't contribute to
the proof of the main theorem.