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Littlewood conjecture

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In mathematics, the Littlewood conjecture is an open problem (as of April 2024) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,

where is the distance to the nearest integer.

Formulation and explanation

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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.

Connection to further conjectures

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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.

Partial results

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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 , it is possible to construct an explicit β such that (α,β) satisfies the conjecture.[5]

See also

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References

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  1. ^ J.W.S. Cassels; H.P.F. Swinnerton-Dyer (1955-06-23). "On the product of three homogeneous linear forms and the indefinite ternary quadratic forms". Philosophical Transactions of the Royal Society A. 248 (940): 73–96. Bibcode:1955RSPTA.248...73C. doi:10.1098/rsta.1955.0010. JSTOR 91633. MR 0070653. S2CID 122708867. Zbl 0065.27905.
  2. ^ Adamczewski & Bugeaud (2010) p.444
  3. ^ M. Einsiedler; A. Katok; E. Lindenstrauss (2006-09-01). "Invariant measures and the set of exceptions to Littlewood's conjecture". Annals of Mathematics. 164 (2): 513–560. arXiv:math.DS/0612721. Bibcode:2006math.....12721E. doi:10.4007/annals.2006.164.513. MR 2247967. S2CID 613883. Zbl 1109.22004.
  4. ^ Adamczewski & Bugeaud (2010) p.445
  5. ^ Adamczewski & Bugeaud (2010) p.446

Further reading

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