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	+}