AI 中文总结
研究最小距离问题，通过构造特定点线对族，结合前人结果解决该问题，给出了与最小距离相关的精确指数结论及方法。
AI 中文摘要
我们证明，对于每个固定的\(\varepsilon>0\)，在\([0,1]^2\)中存在任意大的点线对族\((x_1,\ell_1),\ldots,(x_n,\ell_n)\)，其中\(x_i\in\ell_i\)对所有\(i\)成立，且对于所有\(i\neq j\)，\(\operatorname{dist}(x_i,\ell_j)\geq n^{-2/3 - \varepsilon}\)。结合Cohen、作者和Zakharov之前的结果，解决了最小距离问题。
英文摘要
We show that for every fixed $\varepsilon>0$, there exist arbitrarily large families of point-line pairs $(x_1,\ell_1),\ldots,(x_n,\ell_n)$ in $[0,1]^2$, with $x_i \in \ell_i$ for all $i$, and such that $\operatorname{dist}(x_i,\ell_j)\ge n^{-2/3-\varepsilon}$ for all $i \neq j$. 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 $\varepsilon>0$, 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 $\gtrsim q^{3/2-\varepsilon}$. This disproves a conjecture of Hunter, the author, Verstraëte and Zhang.
Comments14 pages, new Section 5 added
