Let $\lambda\in(0,1)$ be an irrational number such that its continued fraction expansion is bounded (for example, an irrational quadratic number, whose continued fraction is periodic). It is known that the bound $$\left|\lambda-\frac{p}{q}\right|\leq \frac{C}{q^2},$$ where $\frac{p}{q}$ is a convergent (best rational approximation) of $\lambda$, is optimal for some positive constant $C$, namely $$\mathop{\liminf_{p,q\,\in\,\mathbb{Z}}}_{(p,q)=1}\,q |q\lambda-p|>0.$$ Consider now pairs of the form $(\theta,\theta\lambda)$, with $\theta>0$. I would like to understand whether there is a choice of $\theta$ such that $(\theta,\theta\lambda)$ is simultaneously approximabile by rationals with a slightly better bound than $1/q^2$. More precisely, my question is:
does there exists $\theta>0$ such that $$\mathop{\liminf_{p_1,p_2,q\,\in\,\mathbb{Z}}}_{(p_1,q)=(p_2,q)=1}\,\max\{q |q\theta-p_1|,q |q\theta\lambda-p_2|\}=0$$ holds true?
Equivalently, we are asking the following. Let $$W:=\big\{(x_1,x_2)\in \mathbb{R}^+\times \mathbb{R}^+\,\mbox{such that}\mathop{\liminf_{p_1,p_2,q\,\in\,\mathbb{Z}}}_{(p_1,q)=(p_2,q)=1}\,\max_{i=1,2} q |qx_i-p_i|=0\big\},$$ which is a fractal set with Hausdorff dimension $\frac32$. Is it true that $W$ intersects the line passing through $(0,0)$ and $(1,\lambda)$?
A natural candidate for $\theta$ could be a Liouville number, since Liouville numbers have good properties of simultaneous approximation with other irrationals.
It would be already interesting to prove that such a choice of $\theta$ exists for some specific $\lambda$ with bounded continued fraction.
