Copy Theorem

monograph guide
Billiard King homepage
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.