Littlewood conjecture

In mathematics, the Littlewood conjecture is an open problem in Diophantine approximation, proposed by J. E. Littlewood around 1930. It states that for any two real numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$,

${\displaystyle \liminf _{n\to \infty }\ n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0,}$

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 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.

In 1955 Cassels and Swinnerton-Dyer^[1] showed that Littlewood's Conjecture would follow from the following conjecture in the geometry of numbers in the case ${\displaystyle n=3}$:

Conjecture 1: Let L be the product of n linear forms on ${\displaystyle \mathbb {R} ^{n}}$. Suppose ${\displaystyle n\geq 3}$ and L is not a multiple of a form with integer coefficients. Then ${\displaystyle \inf\{|L(x)|\mid x\in \mathbb {Z} ^{n}\setminus \{0\}\}=0}$.

Conjecture 1 is equivalent to the following conjecture concerning the orbits of the diagonal subgroup D on ${\displaystyle SL(n,\mathbb {R} )/SL(n,\mathbb {Z} ),}$ as was essentially noticed by Cassels and Swinnerton-Dyer.

Conjecture 2: Let ${\displaystyle n\geq 3}$. For any ${\displaystyle x\in SL(n,\mathbb {R} )/SL(n,\mathbb {Z} )}$, if the orbit ${\displaystyle Dx}$ is relatively compact, then ${\displaystyle Dx}$ is closed.

This is due to Margulis.^[2] Conjecture 2 is a special case of the following far more general conjecture, also due to Margulis.

Conjecture 3: Let G be a connected Lie group, ${\displaystyle \Gamma }$ a lattice in G, and H a closed connected subgroup generated by ${\displaystyle (Ad_{G},\mathbb {R} )}$-split elements, i.e. all eigenvalues of ${\displaystyle Ad_{G}(g)}$ are real for each generator g. Then for any ${\displaystyle x\in G/\Gamma }$, exactly one of the following holds:

1. ${\displaystyle {\overline {Hx}}}$ is homogeneous, i.e. there is a closed subgroup F of G such that ${\displaystyle {\overline {Hx}}=Fx}$.

1. There exists a closed connected subgroup F of G and a continuous epimorphism ${\displaystyle \phi }$ from F onto a Lie group L such that ${\displaystyle H\subset F}$, ${\displaystyle Fx}$ is closed in ${\displaystyle G/\Gamma }$, ${\displaystyle \phi (F_{x})}$ is closed in L where ${\displaystyle F_{x}}$ is the stabilizer, and ${\displaystyle \phi (H)}$ is a one-parameter subgroup of L containing no non-trivial ${\displaystyle Ad_{L}}$-unipotent elements, i.e. elements g for which 1 is the only eigenvalue of ${\displaystyle Ad_{L}(g)}$.

Borel showed in 1909 that the exceptional set of real pairs (α,β) violating the statement of the conjecture is of Lebesgue measure zero.^[3] Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss have shown^[4] that it must have Hausdorff dimension zero;^[5] 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 (i.e., pairs (α,β) which are individually badly approximable and where 1, α, and β are linearly independent over ${\displaystyle \mathbb {Q} }$) satisfying the conjecture exist: indeed, given a real number α such that ${\displaystyle \inf _{n\geq 1}n\cdot ||n\alpha ||>0}$, it is possible to construct an explicit β such that (α,β) is non-trivial and satisfies the conjecture.^[6]

• Littlewood polynomial

• Adamczewski, Boris; Bugeaud, Yann (2010). "8. Transcendence and diophantine approximation". In Berthé, 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.