Littlewood conjecture

In mathematics, the Littlewood conjecture is a open problem (2004) in diophantine approximation, posed by J. E. Littlewood around 1930. It states that for any two real numbers α and β

lim inf n||α||·||β|| = 0

as n → ∞, where ||x|| denotes the distance from x to an integer.

It is known that this is 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.

Conjecture: for any g in G/Γ such that Dg is relatively compact (in G/Γ), then Dg is closed.

This is turn is a special case of a general conjecture of Margulis on Lie groups.