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
Famous quotes containing the words connection and/or conjectures:
“Self-expression is not enough; experiment is not enough; the recording of special moments or cases is not enough. All of the arts have broken faith or lost connection with their origin and function. They have ceased to be concerned with the legitimate and permanent material of art.”
—Jane Heap (c. 1880–1964)
“After all, it is putting a very high price on one’s conjectures to have a man roasted alive because of them.”
—Michel de Montaigne (1533–1592)