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Freiman’s Constant
I recently asked people on Mastodon "What’s the most surprising fact you’ve learned in the last couple of weeks?" It was a nice way to learn a lot of interesting things. My own biggest recent surprise was this: the number
plays a fundamental role in number theory!
For any irrational we define its Lagrange number to be the supremum of numbers such that
has infinitely many solutions for rationals So, the easier is to approximate by rational numbers, the bigger its Lagrange number is.
Quite famously, the golden ratio has the smallest possible Lagrange number, namely √5. This means it’s as hard as possible to approximate using rational numbers.
The set of all Lagrange numbers is very complicated, and very interesting. But here’s the shocking fact: if
then every real number is a Lagrange number, and is the smallest number with this property!
is called ‘Freiman’s constant’, because he proved this fact. His proof is 100 pages, and I don’t want to read it… not even counting the fact that it’s only available in Russian:
• G. A. Freiman, Diophantine Approximations and the Geometry of Numbers (Markov’s Problem) [in Russian]. Kalinin State University Press, Kalinin, 1975.
There’s a lot more crazy stuff known about the set of all Lagrange numbers, which is called the Lagrange spectrum . As mentioned, the smallest number in the Lagrange spectrum is . The next is The next is The next is In 1879, Markov showed that such numbers form an increasing sequence that converges to 3. They are precisely the Lagrange numbers of numbers whose continued fraction expansion and eventually consists only of 1’s and 2’s and is eventually periodic, like this:
3 is the Lagrange number of every number whose continued fraction expansion eventually consists only of 1’s and 2’s and is not eventually periodic. Above 3 the Lagrange spectrum becomes much richer. It’s a fractal: it has infinitely many gaps, but positive Hausdorff dimension, with the dimension increasing as we move up.
Moreira showed that when we reach the Hausdorff dimension of the Lagrange spectrum hits 1. And as mentioned, Freiman showed that above
the Lagrange spectrum is a half-line. Directly below Freiman’s constant, Freiman showed there is a gap of width roughly
Here is the classic reference in English on this subject:
• Thomas W. Cusick and Mary E. Flahive, The Markov and Lagrange Spectra, AMS Mathematical Surveys and Monographs 30 , AMS, Providence, Rhode Island, 1989.
The Markov spectrum is another set, containing the Lagrange spectrum, and their relationship is very interesting. Here’s a free online reference that reviews all the basics before doing more:
• Carlos Gustavo Moreira, Geometric properties of the Markov and Lagrange spectra, Annals of Mathematics 188 , 145–170.
For a stripped-down account, go here:
• Wikipedia, Markov spectrum.
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This entry was posted on Thursday, May 7th, 2026 at 10:27 pm and is filed under mathematics. You can follow any responses to this entry through the RSS 2.0 feed.<br>You can leave a response, or trackback from your own site.
8 Responses to Freiman’s Constant
John says:
7 May, 2026 at 10:41 pm
‘Carlos Gustavo Moreira, Geometric properties of the Markov and Lagrange spectra’ is at
http://www.arxiv.org/abs/arXiv:1612.05782
not
http://www.arxiv.org/abs/arXiv:1612.0578
Reply
John Baez says:
7 May, 2026 at 11:21 pm
Thanks for catching that!
Reply
Alvaro Riascos says:
8 May, 2026 at 12:38 am
Last reference is taking to a different paper. AR
Reply
John Baez says:
8 May, 2026 at 4:10 pm
I think it’s fine now.
Reply
Allen Knutson says:
8 May, 2026 at 7:34 pm
So what numbers have F as their Lagrange number?
Reply
John Baez says:
8 May, 2026 at 11:41 pm
Good question. In the process of trying to answer this I realized had made a bad mistake, which I’ve tried to fix. I believe the Lagrange spectrum up to 3 works like this:
The smallest number in the Lagrange spectrum is . The next is The next is The next is These numbers form an increasing sequence that converges to 3. They are precisely the Lagrange numbers of numbers whose continued fraction expansion is eventually periodic and eventually consists only of 1’s and 2’s, like this:
3 is the Lagrange number of every number whose continued fraction expansion is not eventually periodic, and eventually consists only of 1’s and 2’s.
Alas, I don’t know a number whose Lagrange number is Freiman’s constant.
Reply
John Baez says:
9 May, 2026 at 10:45 am
I have escalated your question to MathOverflow:
• What is a number whose Lagrange number is Freiman’s constant?
Gro-Tsen made a lot of progress, but apparently hadn’t quite found the right number at the time I’m writing this. He suggested trying
But apparently the Lagrange number of this is not Freiman’s constant but
which is less than Freiman’s constant.
There are reasons to think that the right number...