The Littlewood conjecture

Prove that n‖nα‖‖nβ‖ gets arbitrarily close to 0 for every pair of real numbers α and β.

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1	{
2	"domain": "mathematics",
3	"topics": [
4	"number-theory"
5	],
6	"type": "proof",
7	"source": {
8	"key": "erdos",
9	"name": "Erdős Problems",
10	"url": "https://www.erdosproblems.com/495",
11	"id": "495",
12	"curator": "Thomas Bloom",
13	"poser": "J. E. Littlewood",
14	"status": "open",
15	"as_of": "2026-10-20",
16	"catalog": "0521c19bbd38"
17	},
18	"review": true,
19	"approval": "first-time"
20	}