In Diophantine approximation, the Littlewood conjecture is an open problem concerning the simultaneous approximation of two real numbers by rational numbers with the same denominator. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementAn open problem in Diophantine approximation concerning the simultaneous approximation of two real numbers by rational numbers sharing the same denominator. 1 Proposed YearApproximate; the source states Littlewood proposed the conjecture around 1930, with no exact publication date recorded. Progress Toward ResolutionProved for almost every pair by Patrick Gallagher in 1962. In 2006 Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss proved that the set of possible counterexamples has Hausdorff dimension zero. No general proof or disproof exists. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Littlewood Conjecture (Wikipedia)
Wikimedia FoundationLead section
the Littlewood conjecture is an open problem concerning the simultaneous approximation of two real numbers by rational numbers with the same denominator
Lead section, first sentence
the Littlewood conjecture is an open problem concerning the simultaneous approximation of two real numbers by rational numbers with the same denominator
Lead section, second paragraph
The conjecture was proposed by J. E. Littlewood around 1930 and remains unresolved.
Lead section, third paragraph
For almost every pair, a stronger assertion was proved by Patrick Gallagher in 1962. In 2006, Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss proved that the set of counterexamples has Hausdorff dimension zero, using rigidity of invariant measures for higher-rank diagonal actions on homogeneous spaces.
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