[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?