The Box Theorem

Billiard King homepage
Monograph guide
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.