Problem 52 — Visual explainer (undergrad)
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Problem statement
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Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?\]

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).

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

Benchmarks / known results (from comments)
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- Erd\H{o}s and Szemer\'{e}di proved a lower bound of $\lvert A\rvert^{1+c}$
  for some constant $c>0$, and an upper bound of\[\lvert A\rvert^2
  \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right)\]for
  some constant $c>0$.
- The lower bound has been improved a number of times.
- The current record is\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert
  A\rvert^{\frac{1270}{951}-o(1)}\]due to Bloom (note
  $1270/951=1.33543\cdots$).
- A complete history of sum-product bounds can be found at this webpage.