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The Erdős similarity problem

Show that for every infinite set A of reals some set of positive measure contains no scaled and shifted copy of A.

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1{
2 "domain": "mathematics",
3 "topics": [
4 "analysis"
5 ],
6 "type": "proof",
7 "source": {
8 "key": "erdos",
9 "name": "Erdős Problems",
10 "url": "https://www.erdosproblems.com/120",
11 "id": "120",
12 "curator": "Thomas Bloom",
13 "poser": "Paul Erdős",
14 "status": "open, $100 prize",
15 "as_of": "2026-10-20",
16 "prize": "$100",
17 "catalog": "0521c19bbd38"
18 },
19 "review": true,
20 "approval": "first-time"
21}