Problem 40 — Visual explainer (undergrad)
=========================================

Problem statement
---------------
For what functions $g(N)\to \infty$ is it true that\[\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}\]implies $\limsup 1_A\ast 1_A(n)=\infty$?

Picture
-------
[Given objects + constraints]  --->  [Count / structure]  --->  [Show bound / existence]

Question: what to look at
------------------------
- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Notation (if it appears above)
-----------------------------
- `\gg` / `>>` means "at least a constant times" (for large parameters).

Benchmarks / known results (from comments)
----------------------------------------
- This is a stronger form of the Erd\H{o}s-Tur\'{a}n conjecture [28] (since
  establishing this for any function $g(N)\to \infty$ would imply a positive
  solution to [28]).