Problem 28 — Visual explainer (undergrad)
=========================================

Problem statement
---------------
If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$.

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

Conjecture / Task: 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).

Benchmarks / known results (from comments)
----------------------------------------
- Conjectured by Erd\H{o}s and Tur\'{a}n.
- They also suggest the stronger conjecture that $\limsup 1_A\ast 1_A(n)/\log
  n>0$.
- Another stronger conjecture would be that the hypothesis $\lvert A\cap
  [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.
- Erd\H{o}s and S\'{a}rk\"{o}zy conjectured the stronger version that if
  $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ with $a_n/b_n\to 1$ are
  such that $A+B=\mathbb{N}$ then $\limsup 1_A\ast 1_B(n)=\infty$.