The Markov Constant

Now at last we come to the point: the Markov constant. Let's start by looking at the spectrum of possible values.

We already know that there's one equivalence class with M = R.

After that, there's one equivalence class with M = 2 root 2.

After that, well, it turns out that those two facts represent the start of a pattern. There's one equivalence class for every solution to the Markov equation, and the corresponding values of M increase monotonically with u and converge to 3. Here are the exact equations …

D	= 9u2 − 4
M	= (root D)/u = root(9 − 4/u2)

… and here are the values for the first few solutions.

u	D	M
1	5	2.236068
2	32	2.828427
5	221	2.973214
13	1517	2.996053
29	7565	2.999207
34	10400	2.999423
89	71285	2.999916

And here are some notes.

• “D” stands for “discriminant”, not “determinant”; more about that in a minute.
• Although we take the square root of D, the values of D aren't guaranteed to be square-free. Even in this small sample, 32 is 2⁵ and 10400 is 2⁵ × 5² × 13.
• The values of M converge to 3 very rapidly. In fact, 3 − M is approximately 2/(3u²).

After that, at the limit point M = 3 there are an uncountable number of equivalence classes.

After that, the structure of the spectrum gets complicated. My impression after a quick look around is that it's not even fully understood yet.

In the end, though, the structure gets simple again. Every value M ≥ F is possible, where F is Freiman's constant, approximately 4.528 and exactly this wild number.

(2221564096 + 283748 root 462)/491993569

If you want to find out more about the complicated part, the keyword “Freiman's constant” will take you in the right direction.

To help with that or with any other reading you might want to do, here are definitions of some standard terms that I avoided earlier because they displease me.

• A Markov number is any number that appears in a solution to the Markov equation. Any number that appears in a solution appears in the first position in some solution, so the projection from (u,v,w) to u maps solutions onto Markov numbers. As we discussed, whether that projection is one-to-one is unknown. Consequently, talking about Markov numbers generally just complicates things. When I said there was one equivalence class for every solution to the Markov equation, that was entirely true, although nontrivial, but if I'd said there was one equivalence class for every Markov number, the truth of that would have depended on an unknown.
• A Lagrange number is one of the numbers root(9 − 4/u²). So, there's exactly one per Markov number.
• The Lagrange spectrum is the spectrum of possible values of the Markov constant that we've been talking about.
• The Markov spectrum is a related spectrum that has something to do with quadratic forms. According to the Wikipedia article Markov spectrum, outside the complicated part the Lagrange spectrum and the Markov spectrum are identical, but inside the complicated part the Lagrange spectrum is a proper subset of the Markov spectrum. (That same article is where I found the exact value of F.)

Now that we know the possible values of M, let's investigate which numbers produce which values. Just for M ≤ 3 though, nothing complicated! Here I have something to contribute, one of those obvious things that people don't bother to mention. If a number has an infinite number of coefficients with a_n+1 ≥ 3, then it has an infinite number of approximation qualities with Q_n > a_n+1 ≥ 3, therefore M > 3! Contrapositively, if a number has M ≤ 3, then it only has a finite number of coefficients with a_n+1 ≥ 3, and since we're actually looking for equivalence classes, we can remove a finite prefix that contains those coefficients and require that all coefficients be 1 or 2!

(Could the qualities Q_n get closer and closer to 3 so that M = lim sup Q_n = 3? In general, yes, limits are like that. Here, though, the equation for Q_n makes it impossible. For the qualities Q_n to get closer and closer to 3, the neighboring coefficients a_n and a_n+2 would have to get larger and larger, and that would send M off to infinity.)

So, we want to look at expansions that are all 1s and 2s. We already know about the expansion that's all 1s, so let's see what happens if we add a few 2s, say isolated 2s that get further and further apart. The result? On the one hand, the qualities Q_n where a_n+1 = 2 converge to the following value, approximately 3.236.

[2,1] + [0,1] = (R+3)/2 + (R−1)/2 = R+1

On the other hand, the qualities Q_n where a_n+1 = 1 are bounded below 3 and have no effect on the supremum.

Q_n < a_n+1 + 2 = 3

Therefore, M = R+1.

That didn't give us anything we were looking for, but let's try again. What happens if we add pairs of 2s that get further and further apart? On the one hand, the qualities Q_n where a_n+1 = 2 converge to the following two values … which turn out to be the same.

[2,2,1]	+ [0,1]	= 2 + [0,2,1]	+ [0,1]	= 2 + (3−R)/2	+ (R−1)/2	= 3
[2,1]	+ [0,2,1]	= 2 + [0,1]	+ [0,2,1]	= 2 + (R−1)/2	+ (3−R)/2	= 3

On the other hand, the qualities Q_n where a_n+1 = 1 are bounded below 3 and have no effect on the supremum. Therefore, M = 3!

We were looking for an uncountable number of equivalence classes at M = 3, and that provides them, because there are so many ways for the sizes of the gaps between pairs to increase. For example, we could encode the decimal digits of a real number in the sizes of the gaps. Are there any other expansions with M = 3? I don't know.