Connection To Further Conjectures
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: the implication was shown in 1955 by J. W. S. Cassels and Swinnerton-Dyer. This can be formulated another way, in group-theoretic terms. There is now another conjecture, expected to hold for n ≥ 3: it is stated in terms of G = SLn(R), Γ = SLn(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 in turn is a special case of a general conjecture of Margulis on Lie groups.
Read more about this topic: Littlewood Conjecture
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