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Source. Proposition 11.1, p. 557, of Manfred Einsiedler, Anatole Katok and Elon Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Annals of Mathematics 164 (2006), 513--560, in the edition identified on the source card; the proof is on p. 557.

Statement

Setting (p. 556). For $u,v\in\mathbb R$,

$$\tau_{u,v}=\begin{pmatrix}1&0&0\\ u&1&0\\ v&0&1\end{pmatrix}\operatorname{SL}(3,\mathbb Z),$$

the point of $X=\operatorname{SL}(3,\mathbb R)/\operatorname{SL}(3,\mathbb Z)$ corresponding to the lattice in $\mathbb R^3$ generated by $(1,u,v)$, $(0,1,0)$ and $(0,0,1)$.

Proposition 11.1 (p. 557). The pair $(u,v)$ satisfies

$$\liminf_{n\to\infty}n\langle nu\rangle\langle nv\rangle=0 \qquad (11.1)$$

if and only if the orbit $A^+\tau_{u,v}$ is unbounded, where $A^+$ is the semigroup of matrices $\operatorname{diag}(e^{-r-s},e^{r},e^{s})$ with $r,s\in\mathbb R^+$.

The paper calls the proposition well known, cites Margulis's survey [24, §2] and Starkov's book [46, §30.3] for it, and includes a proof for completeness (pp. 556--557).

Read depth. Claims checked: the setting and the statement were read clause by clause on pp. 556--557, and the proof on p. 557 was read through.

Proof pointer

Page 557. By Mahler's criterion it suffices to show that (11.1) holds exactly when $\inf_{a\in A^+}\delta_{\mathbb R^3}(a\tau_{u,v})=0$, where $\delta_{\mathbb R^3}$ is the length of a shortest nonzero lattice vector. A short vector of $a\tau_{u,v}$ comes from an integer vector $(n,m_1,m_2)$ with $n\ne0$, and the product of its three coordinates is $n(nu+m_1)(nv+m_2)$, which is therefore small. Conversely, given $n$ with $|n(nu+m_1)(nv+m_2)|$ very small, Dirichlet's theorem lets one replace $n$ by a bounded multiple so that both $|nu+m_1|$ and $|nv+m_2|$ are also small, and suitable $r,s>0$ then shrink the vector.

Dependencies

Mahler's compactness criterion and Dirichlet's theorem; no other result of the paper. It supplies Theorem 1.5.

Bears on

• Problem 495: it restates the problem's condition for a pair $(\alpha,\beta)$ as unboundedness of the orbit $A^+\tau_{\alpha,\beta}$; the problem is equivalent to every such orbit being unbounded. The proposition proves nothing toward that.