Inmathematics, theLittlewood conjecture is anopen problem (as of April 2024^[update]) inDiophantine approximation, proposed byJohn Edensor Littlewood around 1930. It states that for any tworeal numbers α and β,

${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}n\Vert n\alpha \Vert \Vert n\beta \Vert =0,}$

where${\displaystyle \Vert x\Vert :=min(|x-\lfloor x\rfloor |,|x-\lceil x\rceil |)}$ is 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

(2α, 2β), (3α, 3β), ... .

For each of these, multiply the distance to the closest line with integerx-coordinate by the distance to the closest line with integery-coordinate. This product will certainly be at most 1/4. The conjecture makes no statement about whether this sequence of values willconverge; it typically does not, in fact. The conjecture states something about thelimit inferior, and says that there is a subsequence for which the distances decay faster than the reciprocal, i.e.

o(1/n)

in thelittle-o notation.

It is known that this would follow from a result in thegeometry of numbers, about the minimum on a non-zerolattice point of a product of three linear forms in three real variables: the implication was shown in 1955 byCassels andSwinnerton-Dyer.^[1] This can be formulated another way, in group-theoretic terms. There is now another conjecture, expected to hold forn ≥ 3: it is stated in terms ofG =SL_n(R), Γ =SL_n(Z), and the subgroupD ofdiagonal matrices inG.

Conjecture: for anyg inG/Γ such thatDg isrelatively compact (inG/Γ), thenDg is closed.

This in turn is a special case of a general conjecture ofMargulis onLie groups.

Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is ofLebesgue measure zero.^[2]Manfred Einsiedler,Anatole Katok andElon Lindenstrauss have shown^[3] that it must haveHausdorff dimension zero;^[4] and in fact is a union of countably manycompact sets ofbox-counting dimension zero. The result was proved by using a measureclassification theorem for diagonalizable actions of higher-rank groups, and anisolation 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${\displaystyle \underset{n\ge 1}{inf}n\cdot ||n\alpha ||>0}$, it is possible to construct an explicit β such that (α,β) satisfies the conjecture.^[5]

• Littlewood polynomial

• Adamczewski, Boris; Bugeaud, Yann (2010). "8. Transcendence and diophantine approximation". InBerthé, Valérie; Rigo, Michael (eds.).Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge:Cambridge University Press. pp. 410–451.ISBN 978-0-521-51597-9.Zbl 1271.11073.

• Akshay Venkatesh (2007-10-29)."The work of Einsiedler, Katok, and Lindenstrauss on the Littlewood conjecture".Bull. Amer. Math. Soc. (N.S.).45 (1):117–134.doi:10.1090/S0273-0979-07-01194-9.MR 2358379.Zbl 1194.11075.

• Diophantine approximation
• Conjectures
• Unsolved problems in mathematics

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