A fractal-like configuration of point-line pairs for the minimal distance problem

We show that for every $n\in \mathbb{N}$ there is a collection of points $p_1,\dots ,p_n$ and lines ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_n$ in the unit square such that for any $i$ we have $p_i\in {\mathrm{\ell }}_i$ and the distance from $p_i$ to any other line ${\mathrm{\ell }}_j$ is at least $c{n}^{\gamma -1}$ for some universal constants $c,\gamma >0$. This is better than a trivial construction by a polynomial factor.