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	}