The Erdős–Turán conjecture on additive bases

Show that if every large integer is a sum of two elements of A, some integers have arbitrarily many such representations.

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1	{
2	"domain": "mathematics",
3	"topics": [
4	"number-theory",
5	"combinatorics"
6	],
7	"type": "proof",
8	"source": {
9	"key": "erdos",
10	"name": "Erdős Problems",
11	"url": "https://www.erdosproblems.com/28",
12	"id": "28",
13	"curator": "Thomas Bloom",
14	"poser": "Paul Erdős and Pál Turán",
15	"status": "open, $500 prize",
16	"as_of": "2026-10-20",
17	"prize": "$500",
18	"catalog": "0521c19bbd38"
19	},
20	"review": true,
21	"approval": "first-time"
22	}