[jagthebeetle]

I never did math past linear algebra/real analysis, so the only concept of sizes I have are countable/uncountable infinities.

Apparently the crux of this proof was showing that "the space of all modular forms with bounded denominators" and "the space of all congruence modular forms" were the same size.

I wonder what kind of expression "size" is here. Presumably not some finite integer, nor one of the simple infinities, since their first step was showing one is "a bit bigger" than the other. I wish this article went into more detail on that.

I definitely remember nerding out about modular forms via Andrew Wiles as a younger self.

[largeluke]

If I understand the intro correctly, the "size" they're referring to is the growth rate of a sequence, where the sequence is counting the dimensions of certain subsets of bounded denominator modular forms.

Let BDMF = bounded denominator modular form. They show congruence BDMFs grow at least N^3, but all BDMFs grow at most N^3*log(N). (The latter bound is the hard part of the proof.) To get the contradiction, they show a hypothetical noncongruence BDMF example would imply additional counterexamples that (just barely) get over the N^3*log(N) bound.

[schoen]

So is this a kind of result akin to the prime number theorem (https://en.wikipedia.org/wiki/Prime_number_theorem) where you have things that are asymptotically guaranteed to be/remain very close to one another?

[bmitc]

If you're interested, the book On the Brink of Paradox covers various sizes of infinity.

https://mitpress.mit.edu/9780262039413/on-the-brink-of-parad...

There was an edX course titled Paradox and Infinity that was normally offered every year around May or June, but it didn't run last year.

https://openlearninglibrary.mit.edu/courses/course-v1:MITx+2...

https://www.youtube.com/watch?v=eoxQXiz9ykQ