>>574 補足

https://en.wikipedia.org/wiki/Littlewood_conjecture
Littlewood conjecture
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References
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?Freely accessible. doi:10.4007/annals.2006.164.513. MR 2247967. Zbl 1109.22004.
(引用終り)

これ、arXiv:mathのリンクから下記に入ると、”Ann. of Math. (2) 164 (2006)”版が公開されているね〜（＾＾
https://arxiv.org/abs/math/0612721
https://arxiv.org/pdf/math/0612721.pdf
Invariant measures and the set of exceptions to Littlewood's conjecture
Manfred Einsiedler, Anatole Katok, Elon Lindenstrauss
(Submitted on 22 Dec 2006)
We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture has Hausdorff dimension zero.
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
Journal reference: Ann. of Math. (2) 164 (2006), no. 2, 513--560
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Part 2. Positive entropy and the set of exceptions to Littlewood’s Conjecture
7. Definitions

11. The set of exceptions to Littlewood’s Conjecture

The following well-known proposition
gives the reduction of Littlewood’s conjecture to the dynamical question which
we studied in Section 10; see also [24, §2] and [46, §30.3]. We include the proof
for completeness.
Proposition 11.1. The tuple (u, v) satisfies
(11.1) liminf n→∞ n ||nu|| ||nv|| = 0,
if and only if the orbit A+τu,v is unbounded where A+ is the semigroup
（略）
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