Littlewood conjecture
In mathematics, the Littlewood conjecture is an open problem (as of 2016[update]) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,
where is here the distance to the nearest integer.
This means the following: take a point (α,β) in the plane, and then consider the sequence of points
For each of these consider the closest lattice point, as determined by multiplying the distance to the closest line with integer x-coordinate by the distance to the closest line with integer y-coordinate. This product will certainly be at most 1/4. The conjecture makes no statement about whether this sequence of values will converge; it typically does not, in fact. The conjecture states something about the limit inferior, and says that there is a subsequence for which the distances decay faster than the reciprocal, i.e.
in the little-o notation.
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 diagonal matrices in G.
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