In Diophantine approximation, the Littlewood conjecture is an open problem concerning the simultaneous approximation of two real numbers by rational numbers with the same denominator. It states that, for every pair of real numbers α {\displaystyle \alpha } and β {\displaystyle \beta } ,
lim inf n → ∞ n ‖ n α ‖ ‖ n β ‖ = 0 , {\displaystyle \liminf _{n\to \infty }n\,\|n\alpha \|\,\|n\beta \|=0,}
where
‖ x ‖ = min m ∈ Z | x − m | {\displaystyle \|x\|=\min _{m\in \mathbb {Z} }|x-m|}
is the distance from x {\displaystyle x} to the nearest integer. The conjecture was proposed by J. E. Littlewood around 1930 and remains unresolved. It holds immediately if either number is rational or, more generally, is not badly approximable. Thus any counterexample would have to consist of two badly approximable numbers such that 1 , α , β {\displaystyle 1,\alpha ,\beta } are linearly independent over Q {\displaystyle \mathbb {Q} } . For almost every pair, a stronger assertion was proved by Patrick Gallagher in 1962. In 2006, Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss proved that the set of counterexamples has Hausdorff dimension zero, using rigidity of invariant measures for higher-rank diagonal actions on homogeneous spaces.
Statement and elementary cases The limit-inferior formulation is equivalent to saying that, for every ε > 0 {\displaystyle \varepsilon >0} , there are infinitely many positive integers n {\displaystyle n} such that
n ‖ n α ‖ ‖ n β ‖ < ε . {\displaystyle n\,\|n\alpha \|\,\|n\beta \|<\varepsilon .}
Geometrically, consider the orbit
n ( α , β ) ( mod Z 2 ) ( n = 1 , 2 , … ) {\displaystyle n(\alpha ,\beta ){\pmod {\mathbb {Z} ^{2}}}\qquad (n=1,2,\ldots )}
on the two-dimensional torus. The two factors ‖ n α ‖ {\displaystyle \|n\alpha \|} and ‖ n β ‖ {\displaystyle \|n\beta \|} are the coordinate distances of this point from the integer lattice. The conjecture says that their product is o ( 1 / n ) {\displaystyle o(1/n)} along a subsequence; it does not require either coordinate distance separately to be o ( 1 / n ) {\displaystyle o(1/n)} . If p {\displaystyle p} and q {\displaystyle q} are nearest integers to n α {\displaystyle n\alpha } and n β {\displaystyle n\beta } , respectively, then the same inequality can be written
| α − p n | | β − q n | < ε n 3 . {\displaystyle \left|\alpha -{\frac {p}{n}}\right|\left|\beta -{\frac {q}{n}}\right|<{\frac {\varepsilon }{n^{3}}}.}
Thus the conjecture asks for unusually good simultaneous rational approximations with a common denominator, measured multiplicatively rather than by the maximum of the two errors. A real number α {\displaystyle \alpha } is badly approximable if
inf n ≥ 1 n ‖ n α ‖ > 0. {\displaystyle \inf _{n\geq 1}n\|n\alpha \|>0.}
For an irrational number, this is equivalent to its continued fraction having bounded partial quotients. If α {\displaystyle \alpha } is not badly approximable, then
0 ≤ n ‖ n α ‖ ‖ n β ‖ ≤ 1 2 n ‖ n α ‖ , {\displaystyle 0\leq n\|n\alpha \|\|n\beta \|\leq {\tfrac {1}{2}}n\|n\alpha \|,}
so the conjecture follows immediately; the same argument applies with α {\displaystyle \alpha } and β {\displaystyle \beta } interchanged. The conjecture also holds when 1 , α , β {\displaystyle 1,\alpha ,\beta } are linearly dependent over Q {\displaystyle \mathbb {Q} } .
Reformulations
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