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 weakening of the Littlewood conjecture
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.


