Littlewood conjecture

The Littlewood conjecture states that for all real numbers α and β, the limit inferior of n‖nα‖‖nβ‖ as n tends to infinity is zero, where ‖x‖ is the distance from x to the nearest integer.

Distance to the nearest integer: The quantity ‖x‖ is the minimum distance from a real number x to an integer. It defines the two approximation errors multiplied in the conjecture.
John Edensor Littlewood: A British mathematician known for major contributions to analysis, number theory, and mathematical physics. He posed the conjecture that now bears his name.
Quadratic irrational: An irrational root of a quadratic polynomial with rational coefficients. Pairs from a common real quadratic field fall within a classical proved case.
Counterexample: A case that disproves a universal mathematical statement. Finding a pair with positive limit inferior would refute the conjecture.
Simultaneous Diophantine approximation: The study of approximating several real numbers by rational data with a shared denominator. The conjecture asks whether both coordinates can be approximated unusually well at once.
Cassels–Swinnerton-Dyer theorem: A theorem establishing the Littlewood conjecture for pairs of real numbers in the same real quadratic field. It is an early landmark partial result for the conjecture.
Perron number: A real algebraic integer greater than one whose other conjugates have smaller absolute value. Special algebraic pairs connected with units illustrate proved instances of the conjecture.
Invariant measure: A measure unchanged by the action of a specified transformation or group. Measure rigidity for diagonal flows is connected to proposed approaches to the conjecture.
Dirichlet's approximation theorem: A theorem guaranteeing bounded-denominator rational approximations to real numbers with errors constrained by the denominator. It supplies the basic one-dimensional approximation principle behind the conjecture.
Real quadratic field: A degree-two extension of the rational numbers embedded in the real numbers. The Cassels–Swinnerton-Dyer result covers pairs drawn from such fields.

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Diophantine approximationRelated: It extends approximation questions from one real number to pairs, where major problems remain.