Littlewood conjecture

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In mathematics, the Littlewood conjecture is an 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>\liminf_{n\to\infty} n\ \|n\alpha\| \ \|n\beta\| = 0,</math>

where ||x|| is the distance from x to the nearest integer. In words, this means the following: take a point (α,β) in the plane, and the consider the sequence of points (2α,2β), (3α,3β), ... and for each of these compute the "distance" to the closest point in the plane with integer coordinates by multiplying the distance to the closest line with integer x-coordinate by the distance to the closest line with integer y-coordinate. This distance will certainly be at most 1/4. While the conjecture makes no statement about whether this sequence of values will converge (it does not, in fact), it says that there is a subsequence for which the distances decay as o(1/n).

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.

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.

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.