Problem 66 — Visual explainer (undergrad)
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Problem statement
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Is there $A\subseteq \mathbb{N}$ such that\[\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}\]exists and is $\neq 0$?

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

Question: what to look at
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- 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)
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- Erd\H{o}s and S\'{a}rk\"{o}zy proved that\[\frac{\lvert 1_A\ast 1_A(n)-\log
  n\rvert}{\sqrt{\log n}}\to 0\]is impossible.
- Erd\H{o}s suggests it may even be true that the $\liminf$ and $\limsup$ of
  $1_A\ast 1_A(n)/\log n$ are always separated by some absolute constant.
- Horv\'{a}th proved that\[\lvert 1_A\ast 1_A(n)-\log n\rvert \leq
  (1-\epsilon)\sqrt{\log n}\]cannot hold for all large $n$.