In mathematics, the Littlewood conjecture is a open problem (as of 2004) in Diophantine approximation, posed by J. E. Littlewood around 1930. It states that for any two real numbers α and β, Math sucks. ... 2004 is a leap year starting on Thursday of the Gregorian calendar. ... In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ... John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

where ||x|| is the distance from x to the nearest integer.

It is known that this would follow from a result in the geometry of numbers, about the minimum on a non-zero lattice point of a product of three linear forms in three real variables. This was shown in 1955 by Cassels and Swinnerton-Dyer. This can be formulated another way, in group-theoretic terms. This is now another conjecture, expected to hold for n ≥ 3: it is stated in terms of G = SL_n(R), Γ = SL_n(Z), and the subgroup D of G of diagonal matrices. In number theory, the geometry of numbers is a topic and method arising from the work of Hermann Minkowski, on the relationship between convex sets and lattices in n-dimensional space. ... See lattice for other meanings of this term, both within and without mathematics. ... 1955 is a common year starting on Saturday of the Gregorian calendar. ... John William Scott Cassels (born July 11, 1922) is a leading British mathematician. ... In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ...

Conjecture: for any g in G/Γ such that Dg is relatively compact (in G/Γ), then Dg is closed. In mathematics, a relatively compact subspace (or relatively compact subset) Y of a topological space X is a subset whose closure is compact. ...

This in turn is a special case of a general conjecture of Margulis on Lie groups. Gregori Aleksandrovich Margulis (first name often given as Gregory, Grigori or Grigory) (born February 24, 1946) is a mathematician known for his far-reaching work on lattices in Lie groups, and the introduction of methods from ergodic theory into diophantine approximation. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

Progress has been made in showing that the exceptional set of real pairs (α,β) violating the statement of the conjecture must be small. Einsiedler, Katok and Lindenstrauss have shown that it must have Hausdorff dimension zero; and in fact is a union of countably many compact sets of box-counting dimension zero. In mathematics, the Hausdorff dimension is an extended non-negative real number, that is in the closed infinite interval [0, ∞], associated to any metric space . ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In fractal geometry, the Minkowski-Bouligand dimension or Minkowski dimension is a way of determining the fractal dimension of a set S in a Euclidean space , or more generally of a metric space (X,d). ...