Littlewood conjecture

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In mathematics, the Littlewood conjecture is an open problem (as of 2011) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β,

\liminf_{n\to\infty} \ n\,\Vert n\alpha\Vert \,\Vert n\beta\Vert = 0,

where \Vert \,\Vert is here the distance to the nearest integer.

Formulation and explanation[edit]

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 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.

o(1/n)

in the little-o notation.

Connection to further conjectures[edit]

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.[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[edit]

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 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 \inf_{n \ge 1} n \cdot || n \alpha || > 0 , it is possible to construct an explicit β such that (α,β) satisfies the conjecture.[5]

See also[edit]

References[edit]

  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. doi:10.1098/rsta.1955.0010. JSTOR 91633. MR 70653. 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. doi:10.4007/annals.2006.164.513. MR 2247967. Zbl 1109.22004. 
  4. ^ Adamczewski & Bugeaud (2010) p.445
  5. ^ Adamczewski & Bugeaud (2010) p.446
  • Adamczewski, Boris; Bugeaud, Yann (2010). "8. Transcendence and diophantine approximation". In Berthé, Valérie; Rigo, Michael. Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications 135. Cambridge: Cambridge University Press. pp. 410–451. ISBN 978-0-521-51597-9. Zbl pre05879512. 

Further reading[edit]