# Improving Diophantine approximation by rescaling

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.

• I believe this is true， depending on whether we can find a number with good continued fraction expansion on the trajectory of rescaling. Commented Feb 16, 2023 at 2:45
• I agree, still it is unclear to me how to find a pair in the rescaling trajectory which has simultaneous good approximation. Do you have any suggestion? Commented Feb 16, 2023 at 9:43
• I think it is hard to imagine how the continued fraction expansion varies on the trajectory, maybe a good understanding of the gauss map $G(x)=\left\{\frac{1}{x}\right\}=\frac{1}{x}-\left\lfloor\frac{1}{x}\right\rfloor$ will help imagine it, I may not be able to make constructive suggestions because I haven't thought about it before. Commented Feb 16, 2023 at 9:52