Freiman’s Constant

Azimuth 2026-05-07

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

$\displaystyle{ F = \frac{2221564096 + 283748 \sqrt{462}}{491993569} }$

plays a fundamental role in number theory!

For any irrational $x,$ we define its Lagrange number to be the supremum of numbers $c$ such that

$\displaystyle{ \left| \frac{p}{q} - x \right| < \frac{1}{cq^2} }$

has infinitely many solutions for rationals $p/q.$ So, the easier $x$ 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. The set of all Lagrange numbers is very complicated, and very interesting.

But here’s the shocking fact: if

$\displaystyle{ F = \frac{2221564096 + 283748 \sqrt{462}}{491993569} }$

then every real number $\ge F$ is a Lagrange number, and $F$ is the smallest number with this property!

$F$ 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.

The proof also appears here:

• Thomas W. Cusick and Mary E. Flahive, The Markov and Lagrange Spectra, AMS Mathematical Surveys and Monographs 30, AMS, Providence, Rhode Island, 1989.

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 $\sqrt{5}$ . The next is $\sqrt{8}.$ The next is $\sqrt{221}/5.$ The next is $\sqrt{1517}/13.$ These numbers form an increasing sequence that converges to 3. These numbers are well-understood: they are the Lagrange numbers of the numbers

$x = a + b \sqrt{n}$

with $a,b \in \mathbb{Q}$ and $n \in \mathbb{N}.$ Such numbers $x$ are precisely those whose continued fraction repeats.

Above 3 the Lagrange spectrum becomes much richer: it has infinitely many gaps, but positive Hausdorff dimension, increasing as we move up. Eventually the Hausdorff dimension hits 1. Later we hit Freiman’s constant,

$F \approx 4.52782956616087914088\dots$

and every number $\ge F$ is in the Lagrange spectrum.

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.