Problem 120 — Visual explainer (undergrad)
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Problem statement
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Let $A\subseteq\mathbb{R}$ be an infinite set. Must there be a set $E\subset \mathbb{R}$ of positive measure which does not contain any set of the shape $aA+b$ for some $a,b\in\mathbb{R}$ and $a\neq 0$?

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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- This is true if $A$ is unbounded or dense in some interval.
- It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a
  countable strictly monotone sequence which converges to $0$.
- Steinhaus has proved this is false whenever $A$ is a finite set.
- This conjecture is known in many special cases (but, for example, it is open
  when $A=\{1,1/2,1/4,\ldots\}$, which is Problem 94 on Green's open problems
  list).