A weakening of the Littlewood conjecture For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^2\rightarrow\mathbb{R}$ by
$$\ell(\alpha,\beta)=\liminf_{n\rightarrow\infty}n\|n\alpha\|\|n\beta\|.$$
The Littlewood conjecture asserts that, for all $(\alpha,\beta)\in\mathbb{R}^2$, we have $\ell (\alpha,\beta)=0$.
Can anyone see how to prove the (seemingly much weaker) statement that, for all $(\alpha,\beta)\in\mathbb{R}$,
$$\inf_{A\in\mathrm{SL}_2(\mathbb{Z})}\ell\left(A\left(
\begin{array}{c}
\alpha\\
\beta\\
\end{array}
\right)\right)=0\quad ?$$
I wouldn't be surprised if this problem has an easy solution, but I haven't yet been able to find one. For the purposes of orientation note that, in the definition of $\ell$ if we replace $\liminf$ by $\inf$, then the problem becomes close to trivial.
 A: I believe so.  Let $K$ be a large positive integer, and $N$ an integer parameter going to infinity. By Dirichlet approximation we can find $n \in [N,2N]$ such that $\|n\alpha\|, \|n\beta\| = O(N^{-1/2})$; we can in fact assume that $\|n\alpha\|, \|n \beta\| \asymp N^{-1/2}$ since otherwise we are done.  For sake of notation let's suppose that the signed fractional parts $\{n\alpha\}, \{n\beta\} \in (-1/2,1/2]$ are positive (and thus comparable to $N^{-1/2}$).  Let $p_j/q_j$ be the last continued fraction approximant to $\{n\alpha\}/\{n\beta\}$ (a quantity comparable to 1) with $q_j \leq K$.  (Note we may assume this ratio is irrational, since Littlewood's conjecture is easy in the commensurate case.) Standard continued fraction theory tells us that
$$ p_{j-1} q_j - p_j q_{j-1} = \pm 1$$
and
$$ p_j - q_j \{n\alpha\}/\{n\beta\} = O(1/q_{j+1})$$
$$ p_{j-1} - q_{j-1} \{n\alpha\}/\{n\beta\} = O(1/q_{j})$$
and so $\| n (p_j \beta - q_j \alpha) \| = O( N^{-1/2}/q_{j+1} ) = O( N^{-1/2}/K)$ and $\| n (p_{j-1} \beta - q_{j-1} \alpha) \| = O( N^{-1/2}/q_j ) = O(N^{-1/2})$.  After pigeonholing we can find (for fixed $K$) an infinite sequence of $N$ such that the $p_j,q_j,p_{j-1},q_{j-1}$ are constant, and thus $\ell( p_j \beta - q_j \alpha, p_{j-1} \beta - q_{j-1} \alpha ) = O(1/K)$.  Letting $K$ to infinity we obtain the claim.
