i.e. could we find a subset $X\subset \mathbb{Q}^2$ such that $\overline{X}=\mathbb{R}^2$ and that for any $x,y\in X$ the distance $|x-y|$ is an irrational number?

I'm considering the following assertion of which I'm not sure :

Given finite rational points $p_1,p_2,\dots,p_n$ , and an open ball $D$ on the plane, there is a rational point $x\in D$ such that $|x-p_i|\in \mathbb{R}\backslash \mathbb{Q}$ for $i=1,2,\dots,n$.

But this assertion accounts to be the following seemingly number-theoretic problem:

Given $n$ pairs $(a_i,b_i) (i=1,2,\dots,n)$ of positive integers such that $a_i^2+b_i^2$ is not square of any integer. Could we find an integer $N\geq 2$ such that the integral pairs $(Na_i+1,Nb_i)$ still satisfy the previous property(i.e. $(Na_i+1)^2+(Nb_i)^2$ is not square of any integer).

BTW the distribution of the Pythagorean triples might help.

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