In mathematics, the Littlewood conjecture is an open problem in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,
\liminfn\toinfty n\Vertn\alpha\Vert\Vertn\beta\Vert=0,
where
\Vertx\Vert:=min(|x-\lfloorx\rfloor|,|x-\lceilx\rceil|)
This means the following: take a point (α, β) in the plane, and then consider the sequence of points
(2α, 2β), (3α, 3β), ... .
For each of these, multiply 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.
o(1/n)
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 Cassels and Swinnerton-Dyer.[1] 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.
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.
Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of Lebesgue measure zero.[2] Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown[3] that it must have Hausdorff dimension zero;[4] and in fact is a union of countably many compact sets of box-counting dimension zero. The result was proved by using a measure classification theorem for diagonalizable actions of higher-rank groups, and an isolation theorem proved by Lindenstrauss and Barak Weiss.
These results imply that non-trivial pairs satisfying the conjecture exist: indeed, given a real number α such that
infnn ⋅ ||n\alpha||>0