The sharp exponent for the minimal distance problem

We show that for every fixed $\epsilon >0$, there exist arbitrarily large families of point-line pairs $(x_1,{\mathrm{\ell }}_1),\dots ,(x_n,{\mathrm{\ell }}_n)$ in $[0,1{]}^2$, with $x_i\in {\mathrm{\ell }}_i$ for all $i$, and such that $\mathrm{dist}(x_i,{\mathrm{\ell }}_j)\ge n^{-2\mathrm{/}3-\epsilon }$ for all $i\mathrm{\ne }j$. Combined with a previous result of Cohen, the author, and Zakharov, this solves the minimal distance problem. The same construction also comes with an unexpected finite field consequence: for every $\epsilon >0$, there exists a set of primes $q$ of positive relative density for which ${\mathbb{F}}_q^2$ contains an induced point-line matching of size $\gtrsim q^{3\mathrm{/}2-\epsilon }$. This disproves a conjecture of Hunter, the author, Verstra"ete and Zhang.