A weakening of the Littlewood conjecture

Question

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.

Answer 1

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.