The Box Theorem

The demo explains the Box Theorem, a
result about the   arithmetic graph  G(p/q)
associated to a rational parameter A=p/q
The applet on the right draws these graphs.
Press the GO button to activate the applet.

 The Basic Result 
Let p/q be a rational with pq odd.
Define V=(q,-p) and
W=(pq/(p+q),(pq/(p+q)+(q-p)/2);

Let R(p/q) be the parallelogram 
with vertices 0; V; W; V+W;
Let L(p/q) denote the line segment
joining the midpoint of the bottom
segment of R(p/q) to the center of R(p/q).

 The Box Theorem  
One period of G(p/q) is contained
in R(p/q), and disjoint from L(p/q). 

The Box Theorem implies that G(p/q) rises
at least q/4 units from the bottom of R(p/q).
This implies that the orbit of (1/q,1) has
diameter at least q/2. 

The Box Theorem is a consequence of the
 Embedding Theorem  and the  Hexagrid Theorem .
The Embedding Theorem says that the 
arithmetic graph is an embedded curve.

Similar results hold when pq is even,
though we haven't given a proof.
In this case, we would let R(p/q) be
the parallelogram with vertices 
-V, V, -V+2W, V+2W  

 father-son decomposition 
In the odd case, there is a finer decomposition,
which we call the Father-Son (FS) decomposition. 
The large R(p/q) contains the union FS(p/q) of two
smaller parallelograms F(p/q) and S(p/q)
such that one period of G(p/q) is contained in FS(p/q).
I think of S as standing for "son's room"
and F as standing for "father's room".
You can toggle the display on the
applet to show this decomposition.

The dividing line between F(p/q) and S(p/q)
has arithmetic significance.  There is a
unique fraction b/a such that a lies in (0,q)
and qb=1+pa.

Our dividing line is parallel to the sides of
R(p/q) and contains the lattice point (a,-b).
The ceiling of S(p/q) is halfway between
the top and bottom of R(p/q).

The portion of G(p/q) in each of S(p/q)
and F(p/q) seems to have approximate
bilateral symmetry.  I don't have a proof
of this.

We deduce the father-son decomposition
from 4 ingredients

1. The  Hexagrid Theorem .
2. The rotational symmetry of the
   extended arithmetic graph.
3. The  Period Copying Theorem .
4. induction on the complexity
   of the rational number.