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. So do all the numbers of the form where is the golden ratio and are rational. 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…